Rational exponents and radicals (Algebra II) · lesson

Fraction exponents are roots

💡 x^(m/n) means take the nth root, then raise it to the m power.

An exponent of 1/2 means square root. 1/3 means cube root. 1/4 means fourth root.

For an exponent like 3/4, the bottom number is the root and the top number is the power. Take the root first. It keeps the numbers small.

🧠 Let's work it out

Find 625^(3/4).

What do you think comes next?

rational exponent: an exponent that is a fraction, like 2/3
cube root: the number that times itself three times gives the value: ∛64 = 4

💡 A negative exponent means 1 over the positive power.

x^(−n) = 1 ÷ x^n. The negative sign does not make the answer negative. It moves the power to the bottom of a fraction.

Work out the positive power first. Then put it under 1.

🧠 Let's work it out

Find 27^(−2/3).

What do you think comes next?

💡 Pull the largest perfect square (or cube) out of the root, then combine like radicals.

√48 = √(16 × 3) = √16 × √3 = 4√3. Look for the largest perfect square that divides the number.

For cube roots, look for a perfect cube: 8, 27, 64, 125. ∛40 = ∛(8 × 5) = 2∛5.

Like radicals have the same root and the same number inside. Add them like like terms: 2√3 + 3√3 = 5√3.

🧠 Let's work it out

Write √12 + √27 in the form a√3.

What do you think comes next?

💡 When you multiply powers with the same base, add the exponents, even when they are fractions.

x^a · x^b = x^(a + b). The rule works the same for fraction exponents.

To add fractions, find a common denominator first.

🧠 Let's work it out

Write x^(1/4) · x^(2/3) as x^n.

What do you think comes next?

💡 Undo x^(m/n) by raising both sides to the flipped power, n/m.

First get the power alone. Divide away any number in front.

Then raise both sides to the reciprocal power. (x^(3/2))^(2/3) = x^1 = x.

🧠 Let's work it out

Solve 5x^(3/2) = 40.

What do you think comes next?