Algebra 1 Exponents Radicals Lesson – Free PDF

Contents

This algebra 1 exponents radicals lesson explains exponent rules, radicals and rational exponents in one printable page, with definitions, a method and a worked example.

A rational exponent writes a root as a power: \(x^{1/n}\) is the \(n\)th root of \(x\), and \(x^{m/n}\) means take the \(n\)th root, then raise to the \(m\)th power. All integer exponent rules still hold for these new exponents, so \(8^{2/3}=4\) and \(\sqrt{72}=6\sqrt{2}\).

Algebra 1Common Core: N-RN.A.1 · N-RN.A.2

Algebra 1 Exponents Radicals Lesson: key facts

Product rule: \(x^{m}\cdot x^{n}=x^{m+n}\); quotient rule: subtract the exponents.
Power of a power: multiply the exponents, \((x^{m})^{n}=x^{mn}\).
\(x^{0}=1\) and \(x^{-n}=\frac{1}{x^{n}}\) for \(x\neq 0\).
\(x^{m/n}=(\sqrt[n]{x})^{m}\): root in the denominator, power in the numerator.
Simplify a square root by pulling out the largest perfect square factor.

What are the rules for integer exponents?

An exponent tells how many times a base is used as a factor. In \(x^{5}\), the base is \(x\) and the exponent is 5. Every rule below comes from counting factors. Assume \(x\neq 0\) and \(y\neq 0\).

Rule	Pattern	Example
Product	\(x^{m}\cdot x^{n}=x^{m+n}\)	\(x^{3}\cdot x^{4}=x^{7}\)
Quotient	\(\frac{x^{m}}{x^{n}}=x^{m-n}\)	\(\frac{x^{9}}{x^{5}}=x^{4}\)
Power of a power	\((x^{m})^{n}=x^{mn}\)	\((x^{2})^{5}=x^{10}\)
Power of a product	\((xy)^{n}=x^{n}y^{n}\)	\((3x)^{2}=9x^{2}\)
Zero exponent	\(x^{0}=1\)	\(7^{0}=1\)
Negative exponent	\(x^{-n}=\frac{1}{x^{n}}\)	\(2^{-3}=\frac{1}{8}\)

Counting factors: \(x^{2}\cdot x^{3}\) has 2 + 3 = 5 factors of \(x\), so it equals \(x^{5}\).

Why is \(x^{0}=1\)? Look at the powers of 2 and keep dividing by 2. The pattern continues past 1 without a break.

Each step down divides by 2, so the pattern forces \(2^{0}=1\), \(2^{-1}=\frac{1}{2}\) and \(2^{-2}=\frac{1}{4}\).

What does a rational exponent mean?

Rational exponent

\(x^{1/n}=\sqrt[n]{x}\), the number whose \(n\)th power is \(x\).

\(x^{m/n}=(\sqrt[n]{x})^{m}=\sqrt[n]{x^{m}}\). The denominator is the root, the numerator is the power.

This definition keeps the power rules working: \((x^{1/2})^{2}=x^{1}\), so \(x^{1/2}\) must be a number that squares to \(x\).

Example. \(8^{2/3}=(\sqrt[3]{8})^{2}=2^{2}=4\). Take the root first; the numbers stay small.

\(8^{2/3}\): the denominator 3 asks for a cube root, the numerator 2 asks for a square. The result is 4.

How do you simplify a square root?

Find the largest perfect square that divides the number under the root.
Split the root: \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\).
Replace the root of the perfect square with its value and keep what is left under the root.

Worked example. \(\sqrt{72}=\sqrt{36\cdot 2}=\sqrt{36}\cdot\sqrt{2}=6\sqrt{2}\). Check: \(6\sqrt{2}\approx 8.49\) and \(\sqrt{72}\approx 8.49\).

Simplifying \(\sqrt{72}\) by pulling out the largest perfect square, 36.

With variables that are positive, an even power comes out whole: \(\sqrt{x^{6}}=x^{3}\). The graph of \(y=x^{1/2}\) shows how slowly square roots grow.

The graph of \(y=x^{1/2}\) passes through the points where \(x\) is a perfect square.

Exponent Rules, Radicals and Rational Exponents: frequently asked questions

What is a number to the zero power?

Any nonzero number to the zero power equals 1. The pattern \(2^{3}=8\), \(2^{2}=4\), \(2^{1}=2\) divides by 2 each step, so the next value is \(2^{0}=1\). The expression \(0^{0}\) is left undefined in Algebra 1.

What does a negative exponent mean?

A negative exponent means a reciprocal, not a negative number. For example, \(5^{-2}=\frac{1}{5^{2}}=\frac{1}{25}\). Move the base across the fraction bar and the exponent becomes positive.

How do you write a radical as a rational exponent?

Put the index of the root in the denominator and the power inside the root in the numerator. So \(\sqrt[3]{x^{5}}=x^{5/3}\) and \(\sqrt{y}=y^{1/2}\).

When do you add exponents and when do you multiply them?

Add exponents when multiplying powers with the same base: \(x^{2}\cdot x^{3}=x^{5}\). Multiply exponents when raising a power to a power: \((x^{2})^{3}=x^{6}\).

How do you know a square root is fully simplified?

No perfect square larger than 1 divides the number under the root, and no fraction or root is left in a denominator. \(5\sqrt{2}\) is simplified, but \(\sqrt{50}\) is not because 25 divides 50.