FOUNDATIONS 05

Powers and Roots

Powers describe repeated multiplication and scaling. Roots reverse powers and recover the quantity that produced a given power.

The main idea

If a square has side length \(x\), its area is \(x^2\). If a cube has side length \(x\), its volume is \(x^3\). Powers therefore arise naturally when dimensions and quantities scale.

\[\boxed{x\longrightarrow x^2\longrightarrow x^3}\]

Roots answer the reverse question. If the area of a square is known, the square root gives its side length.

1. What a power means

For a positive integer \(n\),

\[a^n=\underbrace{a\times a\times\cdots\times a}_{n\text{ factors}}.\]

The number \(a\) is the base and \(n\) is the exponent or index.

\(2^4=2\times2\times2\times2=16\)
\(x^3=x\times x\times x\)

2. Squares and cubes have geometric meaning

\(x^2\) can represent the area of a square of side \(x\). Similarly, \(x^3\) can represent the volume of a cube of side \(x\).

3. Multiplying powers with the same base

When powers with the same base are multiplied, their factors join together:

\[a^ma^n=a^{m+n}.\]

\(x^2x^3\)
\(=(x\cdot x)(x\cdot x\cdot x)\)
\(=x^5\)

4. Dividing powers with the same base

For \(a\ne0\), common factors cancel:

\[\frac{a^m}{a^n}=a^{m-n}.\]

\(\frac{x^5}{x^2}=\frac{x\,x\,x\,x\,x}{x\,x}\)
\(=x^3\)

5. A power raised to another power

Raising \(a^m\) to the power \(n\) creates \(n\) groups of \(m\) factors:

\[(a^m)^n=a^{mn}.\]

\((x^2)^3\)
\(=(x^2)(x^2)(x^2)\)
\(=x^6\)

6. Zero powers

The division law explains why a non-zero number raised to the power zero equals \(1\):

\[\frac{a^m}{a^m}=a^{m-m}=a^0.\]

But the left side equals \(1\), so

\[\boxed{a^0=1\qquad(a\ne0)}.\]

\(\frac{5^3}{5^3}=1\)
but also
\(5^{3-3}=5^0\)
therefore \(5^0=1\)

7. Negative powers

A negative exponent means a reciprocal:

\[a^{-n}=\frac1{a^n},\qquad a\ne0.\]

It does not mean that the value itself is negative.

\(2^{-3}=\frac1{2^3}=\frac18\)
\(x^{-1}=\frac1x\)

8. Roots reverse powers

If \(b^n=a\), then the appropriate principal root reverses that power:

\[\sqrt[n]{a}=b.\]

For example, because \(3^2=9\), the principal square root of \(9\) is \(3\).

\(3\xrightarrow{\text{square}}9\)
\(9\xrightarrow{\text{square root}}3\)

\(2\xrightarrow{\text{cube}}8\)
\(8\xrightarrow{\text{cube root}}2\)

9. Fractional powers

Roots can be written using exponents:

\[a^{1/n}=\sqrt[n]{a},\qquad a^{m/n}=\sqrt[n]{a^m},\]

whenever these expressions are defined in the real numbers.

\(16^{1/2}=\sqrt{16}=4\)
\(8^{1/3}=\sqrt[3]{8}=2\)
\(8^{2/3}=(\sqrt[3]{8})^2=4\)

10. A crucial distinction: \(\sqrt9\) and \(x^2=9\)

The symbol \(\sqrt9\) means the principal, non-negative square root:

\[\sqrt9=3.\]

But solving \(x^2=9\) asks for every real number whose square is \(9\), so there are two solutions.

\(\sqrt9=3\)

but

\(x^2=9\Rightarrow x=\pm3\)

Key takeaways

Powers describe repeated multiplication and scaling. Exponents add when equal bases are multiplied, subtract when they are divided, and multiply when a power is raised to another power. Zero, negative and fractional powers extend the same rules consistently, while roots reverse powers.

Why this matters for calculus

Power functions occur throughout calculus. Expressions such as \(x^2\), \(x^{-1}\) and \(x^{1/2}\) all use the same exponent notation, allowing the power rule for differentiation and integration to apply to a much wider range of functions.