Understand powers and real roots

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Understand powers and real roots

An integer power describes repeated multiplication, not repeated addition. For example, 3⁴ is 3 × 3 × 3 × 3 = 81. Recognising familiar powers of 2, 3, 4 and 5 helps you estimate size and check calculations.

A root reverses a power within a stated convention. The square root √25 is 5; the real cube root of −8 is −2, because cubing −2 gives −8. An even root of a negative number is not real. The principal even root of a positive number is nonnegative.

Keep power notation and solution notation separate. √36 = 6, but solving x² = 36 gives x = 6 or x = −6. The equation asks for all values whose squares fit, while the root symbol selects a particular value.

Worked example: Calculate 4³ and the real cube root of 64.

Multiply 4 × 4 × 4 to obtain 64.

Find the real number whose cube is 64.

Answer: 4³ = 64 and ∛64 = 4.

Check your understanding: Is √49 equal to ±7?

Answer: No; √49 = 7.

Why: The principal square root is nonnegative; x² = 49 is the equation with two solutions.