OCR GCSE Maths · OCR 3

Indices and Surds

50 maths questions with worked answers with answers for OCR 3 Indices and Surds in OCR GCSE Maths (spec J560). Practise with flashcards, a topic quiz, a random quiz across the course, 5 crosswords, or read every question and answer below. Not official OCR material.

All 50 questions and answers

Question 1 · Index laws · OCR 3.01a
Simplify 5⁷ × 5³.
Answer:
Add the indices when multiplying.
5⁷ × 5³ = 5¹⁰.
Question 2 · Index laws · OCR 3.01a
Simplify (2⁴)³.
Answer:
Multiply the indices for a power of a power.
(2⁴)³ = 2¹².
Question 3 · Index laws · OCR 3.01a
What is 7⁰?
Answer:
Any non-zero number to the power 0 is 1.
7⁰ = 1.
Question 4 · Index notation · OCR 3.01a
Write 5 × 5 × 5 × 5 in index form.
Answer:
There are four 5s.
5⁴.
Question 5 · Negative indices · OCR 3.01a
Write 18 as a power of 2.
Answer:
8 = 2³.
18 = 2⁻³.
Question 6 · Powers · OCR 3.01b
Calculate 2⁵.
Answer:
2 × 2 × 2 × 2 × 2.
2⁵ = 32.
Question 7 · Powers · OCR 3.01b
Which is larger, 3⁴ or 4³?
Answer:
3⁴ = 81 and 4³ = 64.
3⁴ is larger.
Question 8 · Powers · OCR 3.01b
Calculate 7².
Answer:
7² means 7 × 7.
7 × 7 = 49.
Question 9 · Powers · OCR 3.01b
Calculate 3³.
Answer:
3³ means 3 × 3 × 3.
3 × 3 × 3 = 27.
Question 10 · Powers and roots · OCR 3.01b
Calculate √(36) + ∛(64).
Answer:
√36 = 6 and ∛64 = 4.
6 + 4 = 10.
Question 11 · Powers and roots · OCR 3.01b
Estimate √30 to 1 decimal place without a calculator.
Answer:
5² = 25 and 6² = 36, so √30 is between 5 and 6; 5.5² = 30.25.
√30 ≈ 5.5.
Question 12 · Negative indices · OCR 3.01c
Evaluate 4⁻².
Answer:
A negative index means the reciprocal.
4⁻² = 14² = 116.
Question 13 · Fractional indices · OCR 3.01c · Higher only
Evaluate 27 to the power 23.
Answer:
Take the cube root first: ∛27 = 3.
Then square: 3² = 9.
Question 14 · Indices · OCR 3.01c
Simplify 3⁴ × 3² as a power of 3.
Answer:
For multiplication with the same base, add the indices.
3⁴ × 3² = 3⁶.
Question 15 · Indices · OCR 3.01c
Calculate 9⁰.
Answer:
Any non-zero number to the power zero equals 1.
Therefore 9⁰ = 1.
Question 16 · Indices · OCR 3.01c · Higher only
Calculate 81¹².
Answer:
A power of one half means the square root.
√81 = 9.
Question 17 · Indices · OCR 3.01c · Higher only
Calculate 27²³.
Answer:
The denominator 3 means cube root; the numerator 2 means square.
The cube root of 27 is 3, and 3² = 9.
Question 18 · Laws of indices · OCR 3.01c
Simplify 6⁸6³.
Answer:
Subtract the indices when dividing.
6⁵.
Question 19 · Laws of indices · OCR 3.01c
Find n if 2ⁿ × 2³ = 2¹⁰.
Answer:
n + 3 = 10.
n = 7.
Question 20 · Laws of indices · OCR 3.01c
Simplify 3⁴ × 3⁻²3⁵.
Answer:
3⁴ × 3⁻² = 3².
3² ÷ 3⁵ = 3⁻³ = 127.
Question 21 · Fractional indices · OCR 3.01c · Higher only
Evaluate 32 to the power 35.
Answer:
Fifth root of 32 = 2.
2³ = 8.
Question 22 · Fractional indices · OCR 3.01c · Higher only
Evaluate (94) to the power −12.
Answer:
(94)^12 = 32.
The negative index gives the reciprocal: 23.
Question 23 · Standard form · OCR 3.02a
Write 340 000 in standard form.
Answer:
Move the decimal point 5 places.
340 000 = 3.4 × 10⁵.
Question 24 · Standard form · OCR 3.02a
Write 0.00072 in standard form.
Answer:
Move the decimal point 4 places right.
0.00072 = 7.2 × 10⁻⁴.
Question 25 · Standard form · OCR 3.02a
Write 5.06 × 10³ as an ordinary number.
Answer:
Multiply 5.06 by 1000.
5060.
Question 26 · Standard form · OCR 3.02a
Calculate (3 × 10⁴) × (5 × 10⁶). Give your answer in standard form.
Answer:
3 × 5 = 15 and 10⁴ × 10⁶ = 10¹⁰.
15 × 10¹⁰ = 1.5 × 10¹¹.
Question 27 · Standard form · OCR 3.02a
Calculate (8 × 10⁹) ÷ (2 × 10³).
Answer:
8 ÷ 2 = 4 and 10⁹ ÷ 10³ = 10⁶.
4 × 10⁶.
Question 28 · Standard form · OCR 3.02a
Light travels 3 × 10⁸ m per second. How far does it travel in 2 × 10³ seconds?
Answer:
Distance = speed × time.
3 × 10⁸ × 2 × 10³ = 6 × 10¹¹ m.
Question 29 · Standard form · OCR 3.02a
Calculate (2 × 10³)(3 × 10⁴) in standard form.
Answer:
Multiply the numbers and add the indices.
(2 × 3) × 10⁷ = 6 × 10⁷.
Question 30 · Standard form · OCR 3.02a
Order from smallest to largest: 9 × 10², 2 × 10³, 1.1 × 10³.
Answer:
Convert to ordinary numbers: 900, 2,000 and 1,100.
Smallest first: 9 × 10², 1.1 × 10³, 2 × 10³.
Question 31 · Standard form · OCR 3.02a
Write 4.7 × 10⁻³ as an ordinary number.
Answer:
Move the decimal point 3 places left.
0.0047.
Question 32 · Standard form · OCR 3.02a
Explain why 23 × 10⁴ is not in standard form.
Answer:
The first number must be at least 1 and less than 10.
In standard form it is 2.3 × 10⁵.
Question 33 · Standard form · OCR 3.02b
Calculate (4 × 10⁵) − (3 × 10⁴). Give your answer in standard form.
Answer:
400 000 − 30 000 = 370 000.
3.7 × 10⁵.
Question 34 · Standard form · OCR 3.02b
Calculate (6 × 10⁻³) × (5 × 10⁷).
Answer:
6 × 5 = 30 and 10⁻³ × 10⁷ = 10⁴.
30 × 10⁴ = 3 × 10⁵.
Question 35 · Exact answers · OCR 3.03a
Calculate 34 of 225 exactly.
Answer:
225 = 125.
34 × 125 = 3620 = 95 = 145.
Question 36 · Exact answers · OCR 3.03a · Higher only
A circle has area 49π cm². Find its exact circumference.
Answer:
πr² = 49π, so r = 7 cm.
Circumference = 2πr = 14π cm.
Question 37 · Exact values · OCR 3.03a
A circle has radius 3 cm. Give its exact circumference.
Answer:
Circumference = 2πr.
Substitute r = 3: 2π × 3 = 6π cm.
Question 38 · Exact calculations · OCR 3.03a
Find the exact area of a circle of radius 5 cm.
Answer:
A = πr² = π × 25.
25π cm².
Question 39 · Exact calculations · OCR 3.03a
Calculate 23 + 16 exactly.
Answer:
46 + 16.
56.
Question 40 · Surds · OCR 3.03b · Higher only
Simplify √72.
Answer:
72 = 36 × 2.
√72 = √36 × √2 = 6√2.
Question 41 · Surds · OCR 3.03b · Higher only
Simplify √12 + √27.
Answer:
√12 = 2√3 and √27 = 3√3.
2√3 + 3√3 = 5√3.
Question 42 · Surds · OCR 3.03b · Higher only
Rationalise the denominator of 6√3.
Answer:
Multiply top and bottom by √3: 6√33.
Simplify: 2√3.
Question 43 · Surds · OCR 3.03b · Higher only
Expand and simplify (3 + √2)(3 − √2).
Answer:
Difference of two squares: 3² − (√2)².
9 − 2 = 7.
Question 44 · Surds · OCR 3.03b · Higher only
Simplify 3√5 + 2√5.
Answer:
The surds have the same root, so add the coefficients.
3√5 + 2√5 = (3 + 2)√5 = 5√5.
Question 45 · Surds · OCR 3.03b · Higher only
Simplify √3 × √12.
Answer:
Combine the roots: √3 × √12 = √36.
√36 = 6.
Question 46 · Surds · OCR 3.03b · Higher only
Rationalise 4√2.
Answer:
Multiply numerator and denominator by √2.
4√2 = 4√22 = 2√2.
Question 47 · Surds · OCR 3.03b · Higher only
Simplify √50 − √8.
Answer:
√50 = 5√2 and √8 = 2√2.
5√2 − 2√2 = 3√2.
Question 48 · Surds · OCR 3.03b · Higher only
Expand and simplify (√5 + 1)².
Answer:
5 + 2√5 + 1.
6 + 2√5.
Question 49 · Surds · OCR 3.03b · Higher only
Rationalise the denominator of 12 − √3.
Answer:
Multiply by 2 + √32 + √3: denominator = 4 − 3 = 1.
2 + √3.
Question 50 · Surds · OCR 3.03b · Higher only
Write √48 in the form a√3.
Answer:
48 = 16 × 3.
4√3.

← OCR 2 Fractions, Decimals and Percentages OCR 4 Approximation and Estimation →

Indices and Surds

All 50 questions and answers

More OCR GCSE Maths topics

Cookies on Knowledge Noodles

Cookie settings