Surds & Advanced Exponents Worksheets

What this worksheet covers

This sheet puts index-law and square-root exercises together. Its optional key prints separately, with worked steps when selected. Choose the same options again to get a new selection of expressions.

An item on this handout asks you to:

• simplify x⁵ × x²
• simplify x⁹ ÷ x³
• simplify (x²)⁵
• evaluate 7⁰ and 3⁻²
• simplify √12
• simplify 2⁴ × 2⁻¹
• write 1/27 as a power of 3

Choosing a level

The rung determines whether an item uses one index law or several, and whether a square root needs simplifying at the end. Warm-Up keeps to a single law with whole-number answers, so 2⁴ × 2⁻¹ is about as far as it goes. Basics practises a single rule with small positive indices before the harder combinations appear. Steady mixes two laws in one line; Mastery adds a negative or fractional index, or a root with a square factor inside.

Set the sheet length anywhere from 3 to 15 questions, using whole numbers. Keep one difficulty for a targeted drill, or choose Mixed difficulties to cycle through the stages from your starting point. The count setting changes the page length, not the complexity of an individual exercise. Pick a short set for a quick check or a longer one for revision.

Write-in lets learners show which law they applied, as well as the simplified result. A–D is useful for a short recall check, but the selected option will not show whether indices were added or multiplied.

A worked example

Look for a square factor in the radical, then use the matching-base rule for the product.

Simplify √12. Look for the largest square inside 12: 12 = 4 × 3, so √12 = √4 × √3 = 2√3. Writing 3√2 would need an 18 under the root, and 12 does not contain one.

Simplify 2⁴ × 2⁻¹. The bases match, so the indices are added: 4 + (−1) = 3, giving 2³ = 8. Multiplying 4 by −1 and reaching 2⁻⁴ is the slip the next section picks up.

Marking with the answer key

These errors come from mixing the product, power-of-a-power and reciprocal rules.

• x⁵ × x² answered x¹⁰. The indices were multiplied. A product of powers adds them, 5 + 2 = 7, so the answer is x⁷.
• (x²)⁵ answered x⁷. This time they were added. A power of a power multiplies: 2 × 5 = 10, so the answer is x¹⁰.
• 3⁻² answered −9. The minus landed on the value instead of the index. A negative index means a reciprocal, 1 ÷ 3² = 1/9.
• √12 answered 3√2. The root was halved rather than split. The largest square inside 12 is 4, not 9, so √12 = 2√3. Check by squaring the proposed result: (2√3)² returns 12.

Printing and using it

Use a short set as a final check before a test: each line asks for one law or one simplification, so a skipped rule is easy to spot. The separate key lets a learner retry a missed line before looking at the method.

Standard leaves one line per item for the index calculation. Large-print separates raised indices from their bases; Compact places up to fifteen short exercises on one side.

Add a Name / Date / Class header when the revision will be collected. Dyslexia-friendly type separates raised indices from their bases. Stretch makes every third item one rung harder on a single-difficulty sheet. Change the heading so it names the rule, for example "Multiplying powers". Keep one blank line beside a surd question, so the square factor can be written before taking it outside the root.

Try two eight-question sheets at one level. Write the law beside each answer on the first, then hide those labels on the second and identify the rule before working. Compare the keys to find which index law still needs review. For any wrong answer, write the original and corrected expressions side by side before moving on.

Keep going

Practise the same laws with instant marking on the surds and exponents ladder, read the surds and exponents lesson for why a negative index becomes a fraction, or print logarithms worksheets, which turn the same powers into a second notation.

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