Work out the value of 9<span class="opg-frac"><span class="opg-num">3</span><span class="opg-den">2</span></span>.
Worked Solution & Answer
(√(9))3 = 33 = 27 B1
Answer: 27
Harder indices — Notes & Worked Examples Show / Hide
Explanation:
For a fractional index mn, the bottom number n is the root and the top number m is the power: am/n = (n√a)m. Always take the root first to keep the numbers small before raising to the power.
A negative index means take the reciprocal (flip the fraction upside down): a−n = 1an and (ab)−n = (ba)n.
Example 1: Find the exact value of (a) 272/3 and (b) 16−3/4.
Solution:
(a) Cube root first, then square: (3√27)2 = 32 = 9
(b) Negative power gives reciprocal, 4th root is 2:
16−3/4 = 1(4√16)3 = 123 = 18
Answer: (a) 9, (b) 18
Example 2: Work out the value of (8125)−2/3 as an improper fraction.
Solution:
Negative sign flips the fraction: (1258)2/3
Cube root top and bottom: (3√1253√8)2 = (52)2
Square top and bottom: 52 ÷ 22 = 254
Answer: 254 (or 614)
Evaluate:
43 + 23
43 + 23
Worked Solution & Answer
A power tells you how many times to multiply the base by itself.
e.g. 43 = 4 × 4 × 4 = 64
72
e.g. 43 = 4 × 4 × 4 = 64
72
Answer: 72
Work out
(a) Write down the value of 100 {}1/2) (1) (b) Find the value of 125 {}2/3) (2)
(a) Write down the value of 100 {}1/2) (1) (b) Find the value of 125 {}2/3) (2)
(a) Write down the value of 100<span class="opg-frac"><span class="opg-num">1</span><span class="opg-den">2</span></span> [1 mark]
(b) Find the value of 125<span class="opg-frac"><span class="opg-num">2</span><span class="opg-den">3</span></span> [2 marks]
Worked Solution & Answer
(a)
√100 = 10 B1
Answer (a): 10
(b)
(∛125)² = 5² = 25 M1 A1
Answer (b): 25
Work out
(a) Work out 251/2 × 81/3 (2)
(b) Find the value of (1/32)3/5 (2)
(a) Work out 251/2 × 81/3 (2)
(b) Find the value of (1/32)3/5 (2)
(a) Work out 25<span class="opg-frac"><span class="opg-num">1</span><span class="opg-den">2</span></span> × 8<span class="opg-frac"><span class="opg-num">1</span><span class="opg-den">3</span></span> [2 marks]
(b) Find the value of (132)<span class="opg-frac"><span class="opg-num">3</span><span class="opg-den">5</span></span> [2 marks]
Worked Solution & Answer
(a)
5 × 2 = 10 M1 A1
Answer (a): 10
(b)
(1/2)³ = 1/8 M1 A1
Answer (b): 18
(a) Write 1/16 in the form 4ⁿ. (1) (b) Work out the value of 8 {}5/3) – 9 {}3/2). (3)
(a) Write 116 in the form 4n where n is an integer. [1 mark]
(b) Work out the value of 8<span class="opg-frac"><span class="opg-num">5</span><span class="opg-den">3</span></span> - 9<span class="opg-frac"><span class="opg-num">3</span><span class="opg-den">2</span></span> [3 marks]
Worked Solution & Answer
(a)
1/16 = 1/4² = 4⁻² B1
Answer (a): 4-2
(b)
(³√8)⁵ – (√9)³ = 2⁵ – 3³ = 32 – 27 = 5 M2 A1
Answer (b): 5
Work out
(a) Find the value of 81 {}-1/2). (2) (b) Find the value of (64/125) {}2/3). (2)
(a) Find the value of 81 {}-1/2). (2) (b) Find the value of (64/125) {}2/3). (2)
(a) Find the value of 81-<span class="opg-frac"><span class="opg-num">1</span><span class="opg-den">2</span></span> [2 marks]
(b) Find the value of (64125)<span class="opg-frac"><span class="opg-num">2</span><span class="opg-den">3</span></span> [2 marks]
Worked Solution & Answer
(a)
1 / √81 = 1/9 M1 A1
Answer (a): 19
(b)
(∛(64/125))² = (4/5)² = 16/25 M1 A1
Answer (b): 1625
(a) Work out the value of (5/4)⁻². [2 marks]
19 (b) Work out the value of (9/100) {}3/2). [2 marks]
19 (b) Work out the value of (9/100) {}3/2). [2 marks]
(a) Work out the value of (54)-2 [2 marks]
Answer
(b) Work out the value of (9100)<span class="opg-frac"><span class="opg-num">3</span><span class="opg-den">2</span></span> [2 marks]
Answer
Worked Solution & Answer
(a)
(5/4)⁻² = (4/5)² = 16/25 (or 0.64) M1 A1
Answer (a): Answer 1625
(b)
(√(9/100))³ = (3/10)³ = 27/1000 (or 0.027) M1 A1
Answer (b): Answer 271000
(a) 2⁷ × 4⁵ = 4ˣ, find x (2) (b) Simplify fully (125p⁶y²⁴) {}2/3) (2)
(a) Calculate the value of x [2 marks]
x =
(b) Simplify fully (125p6y^{24})^{23} [2 marks]
Worked Solution & Answer
(a)
2⁷ × 2¹⁰ = 2²ˣ M1
⇒ 2x = 17
⇒ x = 8.5 A1
⇒ 2x = 17
⇒ x = 8.5 A1
Answer (a): x = 8.5
(b)
125 {}2/3) p {}6×2/3) y {}24×2/3) = 25p⁴y¹⁶ M1 A1
Answer (b): 25p4y^{16}
(a) Value of 36 {}1/2) (1) (b) Value of 23⁰ (1) (c) Work out value of 27 {}-2/3) (2)
(a) Write down the value of 36\tfrac{1}{2} [1 mark]
(b) Write down the value of 230 [1 mark]
(c) Work out the value of 27-\tfrac{2}{3} [2 marks]
Worked Solution & Answer
(a)
√36 = 6 B1
Answer (a): 6
(b)
Any non-zero number to power 0 is 1 B1
Answer (b): 1
(c)
(∛27)⁻² = 3⁻² = 1/9 M1 A1
Answer (c): 19
(a) Work out (9/16) {}-3/2). [3 marks]
19 (b) √125 = 5n. Work out the value of n. [2 marks]
19 (b) √125 = 5n. Work out the value of n. [2 marks]
(a) Work out the value of (916)-<span class="opg-frac"><span class="opg-num">3</span><span class="opg-den">2</span></span> [3 marks]
Answer
(b) √(125) = 5n
Work out the value of n. [2 marks]
Work out the value of n. [2 marks]
n =
Worked Solution & Answer
(a)
(9/16) {}-3/2) = (16/9) {}3/2) = (√(16/9))³ = (4/3)³ = 64/27 M2 A1
Answer (a): Answer 6427
(b)
√125 = √(5³) = 5 {}3/2) M1
⇒ n = 1.5 A1
⇒ n = 1.5 A1
Answer (b): n = 1.5