Algebra 2 Radicals and Rational Exponents Lesson (Free PDF)

Algebra 2 math lesson: Radicals and Rational Exponents – Maths PDF

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This rational exponents lesson for Algebra 2 covers radicals and rational exponents (Common Core N-RN.A.1, N-RN.A.2), with a free PDF to print.

A rational exponent is a root and a power at once: a to the m over n equals the nth root of a, raised to the m. The denominator of the exponent is the index of the root and the numerator is the power. Integer exponent properties still hold, and a negative exponent means a reciprocal.

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

Key facts in this rational exponents lesson

Key facts

a^(1/n) is the nth root of a.
a^(m/n) = (nth root of a)^m: root index = denominator, power = numerator.
a^(-p) = 1 / a^p, also for fractional p.
Product, quotient and power rules work by adding, subtracting and multiplying exponents.
Even roots of negative numbers are not real.

What does a rational exponent mean?

Exponent rules for integers extend to fractions. If \((a^{n})^{m}\) rules keep working, then \((a^{1/2})^{2} = a^{1} = a\), so \(a^{1/2}\) must be a number whose square is \(a\). That is the square root.

Rational exponents

For \(n \ge 2\) and \(a \ge 0\) (any real \(a\) when \(n\) is odd): \(a^{\frac{1}{n}} = \sqrt[n]{a}\). More generally, \(a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^{m} = \sqrt[n]{a^{m}}\), and \(a^{-\frac{m}{n}} = \frac{1}{a^{m/n}}\).

Evaluate 27^(2/3) in two moves: root first, then power.

The denominator of the exponent is the root index, and the numerator is the power. Taking the root first keeps the numbers small: \(27^{2/3} = (\sqrt[3]{27})^{2} = 3^{2} = 9\). You can also take the power first, but then the numbers grow quickly: \(27^{2} = 729\) and \(\sqrt[3]{729} = 9\) gives the same value with more work.

A negative exponent still means a reciprocal, and the reciprocal is handled first.

Negative exponent: reciprocal first, then root and power.

Even roots of negative numbers are not real, so the base must be nonnegative when the root index is even. For example, \((-16)^{1/2}\) is not a real number, while \((-27)^{1/3} = -3\) is.

How do the exponent properties work with fractions?

Property	Rule	Example
Product	\(a^{p} \cdot a^{q} = a^{p+q}\)	\(x^{1/2} \cdot x^{1/3} = x^{5/6}\)
Quotient	\(\frac{a^{p}}{a^{q}} = a^{p-q}\)	\(\frac{x^{3/4}}{x^{1/4}} = x^{1/2}\)
Power of a power	\((a^{p})^{q} = a^{pq}\)	\((x^{2/3})^{3/4} = x^{1/2}\)
Power of a product	\((ab)^{p} = a^{p}b^{p}\)	\((16x^{4})^{1/2} = 4x^{2}\)

Graphs of y = x^(1/2), y = x^(1/3), and y = x. Both root functions rise slowly.

How do you simplify an expression with rational exponents?

Rewrite radicals as rational exponents.
Apply the exponent properties to numbers and variables separately.
Rewrite any negative exponent as a reciprocal.
Evaluate roots of perfect powers and write the final answer without negative exponents.

Example: \(\frac{(8x^{6})^{2/3}}{x^{1/2}}\). First \((8x^{6})^{2/3} = 8^{2/3}x^{4} = 4x^{4}\). Then divide: \(4x^{4 – 1/2} = 4x^{7/2}\).

Four conversions: the root index is the denominator of the exponent.

Rational exponents lesson: questions and answers

What does a fractional exponent mean?

The denominator is the index of a root and the numerator is a power. So 16 to the 3/4 means the fourth root of 16, cubed, which equals 8.

Which comes first, the root or the power?

Either order gives the same value, but taking the root first is easier with whole numbers. For 27 to the 2/3, find the cube root 3, then square it to get 9.

How do you handle a negative rational exponent?

Take the reciprocal first, then evaluate. For 8 to the -2/3, write 1 divided by 8 to the 2/3, which is 1 divided by 4.

Can a rational exponent be applied to a negative base?

Only when the root index is odd. The cube root of -8 is -2, but the square root of -8 is not a real number.

Why does x to the 1/2 equal the square root of x?

The product rule requires x to the 1/2 times x to the 1/2 to equal x to the 1. The only nonnegative number that multiplies by itself to give x is the square root.

How to use this rational exponents lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

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