Surds & Advanced Exponents Practice

What a power actually counts

A power is repeated multiplication: x⁴ means x × x × x × x. The index laws are shortcuts for counting how many copies of a factor are in play, which is why they can be worked out from that definition instead of memorised. A surd is a root that cannot be written as a fraction, such as √2, and simplifying one uses the same counting.

Three results follow straight from the definition. Multiplying powers of the same base adds the counts, dividing subtracts them, and a power of a power multiplies them. A zero index means no factors are left, so the value is 1. A negative index counts how many factors sit below a fraction bar, so a negative power means a reciprocal: 2⁻³ is 1/8, not −8. Both ideas come from the same act of counting, and neither needs a separate rule to memorise. For example, x⁻³ means 1/x³ when x is not zero. The base restriction matters: division by zero is undefined, so negative powers cannot use a zero base.

What this ladder covers

Counting factors is where the ladder begins, and simplifying a surd is where it ends.

• Multiplying powers: keep the base and add the indices, as in x³ × x⁴.
• Dividing powers: keep the base and subtract the indices, as in x⁷ ÷ x⁴.
• A power of a power: multiply the two indices, as in (x³)⁴.
• Zero and negative indices: show why x⁰ = 1, and rewrite 2⁻³ as a fraction.
• Simplifying surds: pull the largest perfect square out of a root, so √8 becomes 2√2.
• Exact against decimal: keep a surd exact unless a rounded decimal is asked for.

Worked examples

Simplify x³ × x⁴. Three factors multiplied by four more gives seven in total, so the answer is x⁷. The indices add because the multiplication is counting copies of the same base.

Simplify x⁷ ÷ x⁴. Seven factors with four cancelled leaves three, so the answer is x³. Subtraction of the indices is the short form of that cancellation.

Simplify (x³)⁴. Four brackets, each holding three factors, make twelve: the answer is x¹². A power of a power multiplies, because the counts are grouped rather than added.

Evaluate 5⁰ and 2⁻³. Any non-zero base to the power 0 is 1, since no factors remain. For 2⁻³, the negative index puts three factors underneath: 1 ÷ 2³ = 1 ÷ 8 = 1/8.

Simplify √8. Split 8 into 4 × 2, because 4 is a perfect square: √8 = √(4 × 2) = √4 × √2 = 2√2. A root of a product is the product of the roots, which is the move that simplifies every surd.

Simplify √50. Since 50 = 25 × 2 and 25 is a square, √50 = √25 × √2 = 5√2. Look for the largest square factor first, because an incomplete split leaves a surd that can still be simplified.

Simplify 3² × 3⁻². Add the indices: 2 + (−2) = 0, so the expression is 3⁰ = 1. Any number multiplied by its own reciprocal gives 1, which is the same fact in another form.

Mistakes worth watching for

With the same base, powers are counted rather than multiplied.

• x³ × x⁴ = x¹². Multiplying the indices belongs to a power of a power. Here the factors sit side by side, so the counts add and the answer is x⁷.
• (x³)⁴ = x⁷. This is the reverse mix-up: adding when the indices are multiplied. Four brackets of three factors make twelve, so x¹² is right.
• 2⁻³ = −8. A negative index means a reciprocal, not a negative answer. 2⁻³ is 1/8, while −8 would be (−2)³, a different expression entirely.
• √8 = 2 + √2. The root splits across multiplication, not addition. √8 = 2√2, about 2.83, while 2 + √2 is about 3.41.
• Rounding a surd too early. Writing √8 as 2.83 and then squaring it loses the exact value. Keep the surd until the final line, and give a decimal only if the question asks for one.
• Using a negative power with a zero base. 0⁻² would mean 1/0², which is undefined. Negative indices require a non-zero base because they place the factors in the denominator.

How the ladder works

The Warm-Up holds five questions and the higher rungs hold eight or nine. An 80% first-try score opens the next rung, though none of them start locked, and every retry comes with different numbers.

Keep going

Print a mixed set from the surds and exponents worksheet generator, use the general calculator for the powers you are still unsure of, or carry the same counting into logarithms, where an index becomes the answer.

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