Algebra
Exponents and scientific notation
Rational exponents
20 practice questions
2 video lessons
Theory + worked examples
Theory
A rational exponent is a fraction β the denominator is a root, the numerator a power:
\[b^{1/n}=\sqrt[n]{b},\qquad b^{m/n}=\sqrt[n]{b^m}.\]
Denominator = root, numerator = power. Take the root first to keep numbers small.
Rational exponents as roots and powers.
Worked examples.
The definition:
\[b^{1/n}=\sqrt[n]{b},\qquad b^{m/n}=\left(\sqrt[n]{b}\right)^m\]
The usual exponent rules still apply.
How to evaluate
- Read the denominator as the root.
- Read the numerator as the power.
- Take the root first.
- Then apply the power.
Example 1 β Square root exponent
Evaluate \(9^{1/2}\).
Solution
The \(1/2\) power is a square root.
| \(9^{1/2}\) | \(=\) | \(\sqrt9=3\) |
Example 2 β Cube root exponent
Evaluate \(8^{1/3}\).
Solution
The \(1/3\) power is a cube root.
| \(8^{1/3}\) | \(=\) | \(\sqrt[3]{8}=2\) |
Example 3 β Power and root
Evaluate \(4^{3/2}\).
Solution
Take the root, then the power.
| \(4^{3/2}\) | \(=\) | \((\sqrt4)^3=2^3=8\) |
Example 4 β Radical to exponent
Write \(\sqrt{x}\) with a rational exponent.
Solution
A square root is the \(1/2\) power.
| \(\sqrt{x}\) | \(=\) | \(x^{1/2}\) |
Common pitfalls
Denominator is the root, not the power.
Take the root first to avoid large numbers.
\(b^{1/2}=\sqrt b\), not \(\dfrac b2\).
Frequently asked questions
What does \(b^{1/2}\) mean?
The square root of \(b\).
What does \(b^{m/n}\) mean?
The \(n\)-th root of \(b^m\).
Evaluate \(27^{1/3}\).
\(3\).
Write \(\sqrt[3]{x}\) with an exponent.
\(x^{1/3}\).
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