Integer Exponents, Roots and Standard Form

Q: What is a power, and what do its base and exponent represent?

A power represents repeated multiplication in the form \(b^n\), where the base \(b\) is the number being multiplied and the exponent \(n\) indicates how many times the base is used as a factor.

Q: How does the placement of brackets affect evaluating a power with a negative base?

With brackets, the exponent applies to both the negative sign and the number, yielding a positive result for even exponents and a negative result for odd exponents; without brackets, the exponent applies only to the number.

Q: What is the rule for multiplying powers that have the same base?

Keep the base and add the exponents: \(a^m \times a^n = a^{m+n}\).

Q: What is the rule for dividing powers with the same base?

Keep the base and subtract the exponent of the denominator from the exponent of the numerator: \(\frac{a^m}{a^n} = a^{m-n}\) (where \(a \neq 0\)).

Q: What is the rule when raising a power to another exponent?

Multiply the exponents together: \((a^m)^n = a^{m \times n}\).

Q: How are powers of products and quotients simplified?

Apply the exponent to every factor in the product or both terms in the quotient: \((ab)^n = a^n b^n\) and \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\) (for \(b \neq 0\)).

Q: How is a negative exponent evaluated?

A negative exponent represents the reciprocal of the base raised to the positive power: \(a^{-n} = \frac{1}{a^n}\) (where \(a \neq 0\)).

Q: What is the difference between even and odd roots regarding negative inputs?

Even roots have no real solutions for negative numbers, whereas odd roots can accept negative inputs and yield negative results.

Q: What is the definition of a number written in standard form?

A number in standard form is written as \(A \times 10^k\), where \(1 \le A < 10\) and \(k\) is an integer (\(k \in \mathbb{Z}\)).

Practice

Mathematics · IB Middle Years Programme (MYP) · 9 min read · Updated: 22 Oct 2026

Integer Exponents, Roots and Standard Form

Welcome to your study notes on Integer Exponents, Roots and Standard Form! Have you ever wondered how scientists write the distance from the Earth to the Sun without filling an entire line with zeros? Or how biologists describe the size of a microscopic bacterium? The answer lies in the power of exponents and roots. In this chapter, you will discover how repeated multiplication creates patterns, how negative powers describe tiny fractions, and how standard form allows us to write enormous and tiny numbers with elegance and precision.

Note: In this chapter, we focus on integer exponents (positive whole numbers, zero, and negative whole numbers). For fractional exponents and logarithms, see the extended chapter on Fractional Exponents, Logarithms and Number Bases (Extended).

---

1. Understanding Powers and Integer Exponents

What is a Power?

A power is a shorthand way of writing repeated multiplication. A power has two key components: the base and the exponent (also called an index or power).

In the expression \(b^n\):
• The base (\(b\)) is the number being multiplied.
• The exponent (\(n\)) tells us how many times the base is used as a factor.

For example:
\(3^4 = 3 \times 3 \times 3 \times 3 = 81\)
Here, \(3\) is the base, \(4\) is the exponent, and \(81\) is the evaluated value.

Handling Negative Bases

One of the easiest places to make a mistake in an exam is forgetting brackets around a negative base:

• \((-2)^4 = (-2) \times (-2) \times (-2) \times (-2) = +16\) (An even exponent makes a negative base positive).
• \((-2)^3 = (-2) \times (-2) \times (-2) = -8\) (An odd exponent keeps a negative base negative).
• \(-2^4 = -(2 \times 2 \times 2 \times 2) = -16\) (Without brackets, the exponent applies only to the \(2\), not the minus sign!).

Quick Tip: Always use brackets on your calculator when evaluating powers with negative bases, like \((-5)^2\)!

---

2. The Laws of Indices (Integer Exponent Rules)

When working with powers that share the same base, we can use general rules to simplify them quickly. These rules are essential for solving problems across MYP Criterion A (Knowing and Understanding).

Law 1: Multiplying Powers with the Same Base

When multiplying powers with the identical base, add the exponents:
\(a^m \times a^n = a^{m+n}\)

Why does this work?
\(2^3 \times 2^2 = (2 \times 2 \times 2) \times (2 \times 2) = 2 \times 2 \times 2 \times 2 \times 2 = 2^5\)
Notice that \(3 + 2 = 5\).

Law 2: Dividing Powers with the Same Base

When dividing powers with the identical base, subtract the exponents:
\(\frac{a^m}{a^n} = a^{m-n}\) (where \(a \neq 0\))

Example:
\(\frac{5^6}{5^2} = 5^{6-2} = 5^4 = 625\)

Law 3: Power of a Power

When raising a power to another exponent, multiply the exponents:
\((a^m)^n = a^{m \times n}\)

Example:
\((x^4)^3 = x^{4 \times 3} = x^{12}\)

Law 4: Power of a Product

When a product is raised to an exponent, apply the exponent to each factor inside:
\((ab)^n = a^n b^n\)

Example:
\((3y)^3 = 3^3 \times y^3 = 27y^3\)

Law 5: Power of a Quotient (Fraction)

When a fraction is raised to an exponent, apply the exponent to both numerator and denominator:
\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\) (where \(b \neq 0\))

Example:
\(\left(\frac{2}{5}\right)^3 = \frac{2^3}{5^3} = \frac{8}{125}\)

Key Takeaway: Exponent rules only apply when the bases are identical. You cannot combine \(2^3 \times 3^4\) into a single base using these laws!

---

3. Zero and Negative Exponents

In MYP Criterion B (Investigating Patterns), you are often asked to discover a rule by extending a pattern. Let’s investigate powers of \(2\):

\(2^4 = 16\)
\(2^3 = 8\) (divide by \(2\))
\(2^2 = 4\) (divide by \(2\))
\(2^1 = 2\) (divide by \(2\))

What happens if we continue dividing by \(2\)?

The Zero Exponent Rule

\(2^0 = 2 \div 2 = 1\)

Any non-zero base raised to the power of zero equals \(1\):
\(a^0 = 1\) (for any \(a \neq 0\))

Examples:
• \(7^0 = 1\)
• \((-100)^0 = 1\)
• \((4x^2y)^0 = 1\)

Negative Exponents

Continuing the pattern further down:
\(2^{-1} = 1 \div 2 = \frac{1}{2} = \frac{1}{2^1}\)
\(2^{-2} = \frac{1}{2} \div 2 = \frac{1}{4} = \frac{1}{2^2}\)
\(2^{-3} = \frac{1}{4} \div 2 = \frac{1}{8} = \frac{1}{2^3}\)

A negative exponent indicates a reciprocal, not a negative value!
\(a^{-n} = \frac{1}{a^n}\) and \(\frac{1}{a^{-n}} = a^n\) (where \(a \neq 0\))

Examples:
• \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\)
• \(5^{-1} = \frac{1}{5}\)
• \(\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2 = \frac{9}{4}\)

Common Mistake to Avoid: A negative exponent does NOT make the number negative. For instance, \(4^{-2} \neq -16\). Instead, \(4^{-2} = \frac{1}{4^2} = \frac{1}{16}\).

---

4. Roots and Radicals

What is a Root?

A root is the inverse (opposite) operation of raising a base to a power.

Square Roots (\(\sqrt{x}\))

The square root of a number \(x\) is a number that, when multiplied by itself, gives \(x\).
Because \(6 \times 6 = 36\), we write \(\sqrt{36} = 6\).
The symbol \(\sqrt{\phantom{x}}\) is called the radical sign, and by convention, it refers to the principal (positive) square root.

Note on real numbers: You cannot find the square root of a negative real number because no real number multiplied by itself gives a negative result (\(\sqrt{-25}\) has no real solution).

Cube Roots (\(\sqrt[3]{x}\))

The cube root of a number \(x\) is a number that, when used as a factor three times, gives \(x\).
Because \(4 \times 4 \times 4 = 64\), we write \(\sqrt[3]{64} = 4\).
Unlike square roots, you can take the cube root of a negative number:
\(\sqrt[3]{-27} = -3\) because \((-3) \times (-3) \times (-3) = -27\).

Higher Roots

The \(n^{\text{th}}\) root of a number is written as \(\sqrt[n]{x}\), where \(n\) is the index of the root.
• \(\sqrt[4]{16} = 2\) because \(2^4 = 16\)
• \(\sqrt[5]{32} = 2\) because \(2^5 = 32\)

Key Takeaway: Even roots (like square roots and 4th roots) require non-negative inputs in the real number system. Odd roots (like cube roots and 5th roots) can take negative inputs.

---

5. Standard Form (Scientific Notation)

In science, geography, and engineering, we frequently encounter quantities that are extremely large (e.g., the mass of a planet) or extremely small (e.g., the mass of an atom). Standard form (also called scientific notation) provides a compact, consistent way to represent these quantities using powers of 10.

The Definition of Standard Form

A number written in standard form is written as:
\(A \times 10^k\)
where:
• \(1 \le A < 10\) (meaning \(A\) must be at least \(1\), but strictly less than \(10\))
• \(k\) is an integer (\(k \in \mathbb{Z}\): positive, negative, or zero)

Converting Large Numbers to Standard Form (\(k > 0\))

For large numbers, move the decimal point to the left until there is only one non-zero digit before the decimal point. The number of places you moved is the positive exponent \(k\).

Step-by-Step Example: Write \(45\,000\,000\) in standard form.
1. Place the decimal after the first non-zero digit: \(4.5\).
2. Count how many places the decimal moved from the end: \(7\) places left.
3. Write with the power of 10: \(4.5 \times 10^7\).

Converting Small Numbers to Standard Form (\(k < 0\))

For decimals smaller than 1, move the decimal point to the right until it is right after the first non-zero digit. The number of places you moved is the negative exponent \(k\).

Step-by-Step Example: Write \(0.000\,082\) in standard form.
1. Place the decimal after the first non-zero digit: \(8.2\).
2. Count how many places the decimal moved: \(5\) places right.
3. Write with the negative power of 10: \(8.2 \times 10^{-5}\).

Converting from Standard Form to Ordinary Numbers

• If the exponent is positive, move the decimal point to the right (making the number bigger):
\(6.03 \times 10^4 = 60\,300\)
• If the exponent is negative, move the decimal point to the left (making the number smaller):
\(3.7 \times 10^{-3} = 0.0037\)

Check Yourself: Is it in Standard Form?
• \(12.5 \times 10^4\) — No! \(12.5\) is not between \(1\) and \(10\). Correct form: \(1.25 \times 10^5\).
• \(0.82 \times 10^{-3}\) — No! \(0.82\) is less than \(1\). Correct form: \(8.2 \times 10^{-4}\).
• \(7.0 \times 10^0\) — Yes! \(7.0\) is between \(1\) and \(10\), and \(10^0 = 1\).

---

6. Calculating with Standard Form

MYP Criterion A tasks often require you to perform operations with standard form numbers without converting them into long ordinary numbers first.

Multiplication

Multiply the number parts (\(A\)) and add the exponents using index laws:

Example: Calculate \((3 \times 10^4) \times (5 \times 10^6)\)
1. Multiply the coefficients: \(3 \times 5 = 15\)
2. Multiply the powers of 10: \(10^4 \times 10^6 = 10^{4+6} = 10^{10}\)
3. Combine: \(15 \times 10^{10}\)
4. Adjust to standard form (since \(15 \ge 10\)): \(15 = 1.5 \times 10^1\), so \(1.5 \times 10^1 \times 10^{10} = 1.5 \times 10^{11}\).

Division

Divide the number parts (\(A\)) and subtract the exponents:

Example: Calculate \(\frac{2.4 \times 10^3}{8 \times 10^7}\)
1. Divide the coefficients: \(\frac{2.4}{8} = 0.3\)
2. Divide the powers of 10: \(10^{3-7} = 10^{-4}\)
3. Combine: \(0.3 \times 10^{-4}\)
4. Adjust to standard form: \(0.3 = 3 \times 10^{-1}\), so \(3 \times 10^{-1} \times 10^{-4} = 3 \times 10^{-5}\).

Addition and Subtraction

To add or subtract, the powers of 10 must be the same before adding or subtracting the coefficients.

Example: Calculate \((4.2 \times 10^5) + (3.1 \times 10^4)\)
1. Rewrite the smaller power to match the larger power:
\(3.1 \times 10^4 = 0.31 \times 10^5\)
2. Factor out the common power and add:
\((4.2 + 0.31) \times 10^5 = 4.51 \times 10^5\)

---

7. Real-World Applications & MYP Criterion Links

In MYP Mathematics, learning is not just about calculations; it connects directly to authentic global contexts and assessment criteria:

• Criterion A (Knowing and Understanding): Accurately evaluating expressions involving negative exponents, applying exponent laws algebraically, and calculating with standard form.
• Criterion B (Investigating Patterns): Looking at numerical tables of powers, describing general rules for \(a^0\) and \(a^{-n}\), and verifying patterns with test values.
• Criterion C (Communicating): Showing full working step-by-step, using correct mathematical symbols (\(\sqrt{\phantom{x}}\), \(\times 10^k\)), and expressing solutions clearly with appropriate units.
• Criterion D (Applying Mathematics in Real-Life Contexts): Using standard form in scientific contexts (such as measuring distances in astrophysics, computer memory in gigabytes (\(10^9\)), or wavelengths of light in nanometres (\(10^{-9}\))), and commenting on whether your calculated answer makes practical sense.

---

8. Chapter Summary & Quick Review

Before moving on to the next chapter, ensure you are comfortable with these core principles:

• Multiplication Rule: \(a^m \times a^n = a^{m+n}\)
• Division Rule: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a Power: \((a^m)^n = a^{m \times n}\)
• Zero Power: \(a^0 = 1\) (where \(a \neq 0\))
• Negative Exponent: \(a^{-n} = \frac{1}{a^n}\)
• Square & Cube Roots: \(\sqrt{a}\) and \(\sqrt[3]{a}\) are the inverse operations of squaring and cubing.
• Standard Form: Always written as \(A \times 10^k\) where \(1 \le A < 10\) and \(k\) is an integer.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is a power, and what do its base and exponent represent?

A power represents repeated multiplication in the form \(b^n\), where the base \(b\) is the number being multiplied and the exponent \(n\) indicates how many times the base is used as a factor.

How does the placement of brackets affect evaluating a power with a negative base?

With brackets, the exponent applies to both the negative sign and the number, yielding a positive result for even exponents and a negative result for odd exponents; without brackets, the exponent applies only to the number.

What is the rule for multiplying powers that have the same base?

Keep the base and add the exponents: \(a^m \times a^n = a^{m+n}\).

What is the rule for dividing powers with the same base?

Keep the base and subtract the exponent of the denominator from the exponent of the numerator: \(\frac{a^m}{a^n} = a^{m-n}\) (where \(a \neq 0\)).

What is the rule when raising a power to another exponent?

Multiply the exponents together: \((a^m)^n = a^{m \times n}\).

How are powers of products and quotients simplified?

Apply the exponent to every factor in the product or both terms in the quotient: \((ab)^n = a^n b^n\) and \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\) (for \(b \neq 0\)).

What is the value of any non-zero base raised to the power of zero?

Any non-zero base raised to the power of zero equals \(1\): \(a^0 = 1\) (where \(a \neq 0\)).

How is a negative exponent evaluated?

A negative exponent represents the reciprocal of the base raised to the positive power: \(a^{-n} = \frac{1}{a^n}\) (where \(a \neq 0\)).

What is the difference between even and odd roots regarding negative inputs?

Even roots have no real solutions for negative numbers, whereas odd roots can accept negative inputs and yield negative results.

What is the definition of a number written in standard form?

A number in standard form is written as \(A \times 10^k\), where \(1 \le A < 10\) and \(k\) is an integer (\(k \in \mathbb{Z}\)).

How do you convert numbers between ordinary decimal form and standard form?

Place the decimal after the first non-zero digit to get \(A\), and let \(k\) be the number of places moved (positive if the original number is \(\ge 10\), negative if it is \(< 1\)).

How do you multiply or divide numbers written in standard form?

Multiply or divide the numerical coefficients \(A\), apply index laws to the powers of 10, and adjust the final answer so that \(1 \le A < 10\).

How do you add or subtract numbers expressed in standard form?

Rewrite the terms so that they share the same power of 10, add or subtract the coefficients, and adjust back to standard form if needed.

Ready to test yourself?

8 practice questions on this topic, with worked solutions. Multiple choice is marked as you go. No account needed.

Practise This Topic

More Mathematics chapters

* The content provided by thinka is generated by AI and may not always be accurate or up-to-date. Please use it as a supplementary resource and verify with official materials.

Put These Notes into Practice

Reading the notes is a great start. Now practise with unlimited AI-generated questions and get instant feedback. 100,000+ students are already improving their grades.

Start Practising Now View Pricing

Done reading? Test yourself with AI practice questions

Practise This Topic Now