Calculating with roots and powers

Question 1

What does a power (exponent) mean in maths?

Answer 1

A power tells you how many times to multiply the base number by itself.

Example: 3² = 3 × 3 = 9; 2⁴ = 2 × 2 × 2 × 2 = 16.

Question 2

What does a negative power mean?

Answer 2

A negative power flips the number — it’s the reciprocal.

Example: a⁻ⁿ = 1 / aⁿ; 5⁻² = 1 / 25.

Question 3

What does a fractional power mean?

Answer 3

It’s a combo of a root and a power.

a^(1/n) = ⁿ√a; a^(m/n) = ⁿ√(a^m) or (ⁿ√a)^m; Example: 8^(1/3) = ∛8 = 2.

Question 4

What is the square root and cube root?

Answer 4

√x = square root; ∛x = cube root.

They reverse squaring or cubing; only even roots need positive results.

Question 5

What is a surd?

Answer 5

A surd is an irrational root — it can’t be simplified to a nice rational number.

Examples: √2, √3, and √5 are surds.

Question 6

How do you simplify surds?

Answer 6

Break it down using square factors.

Example: √72 = √(36 × 2) = 6√2.

Question 7

How do you multiply and divide surds?

Answer 7

Multiply: √a × √b = √(ab); Divide: √a / √b = √(a/b).

Example: √2 × √3 = √6; √8 / √2 = 2.

Question 8

What are the laws of indices (aka exponent rules)?

Answer 8

aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ / aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1 / aⁿ; a^(1/n) = ⁿ√a.

These rules are fundamental for manipulating exponents.

Question 9

Simplify (81)^(3/4)

Answer 9

Step 1: Rewrite as root + power = (⁴√81)³.

⁴√81 = 3; Result: 3³ = 27.

Question 10

Simplify √50 + √18

Answer 10

Step 1: √50 = 5√2; √18 = 3√2; Step 2: 5√2 + 3√2 = 8√2.

Combine like terms after simplification.