Surds & Indices Questions and Answers

Q: Find the value of \(\left(\dfrac{81}{16}\right)^{-3/4}\).

Correct answer: 8/27Recognise the base as a fourth power.\[\frac{81}{16} = \left(\frac{3}{2}\right)^4\]\[\left(\left(\frac{3}{2}\right)^4\right)^{-3/4} = \left(\frac{3}{2}\right)^{-3} = \left(\frac{2}{3}\right)^3 = \frac{8}{27}\]The negative sign in the index inverts the fraction; the denominator 4 takes the fourth root and the numerator 3 cubes the result.

Q: If \(m\) and \(n\) are whole numbers such that \(m^{n}=121\), the value of \((m-1)^{n+1}\) is:

\( 121 = 11^{2} \), so \( m = 11 \) and \( n = 2 \).\[ (m-1)^{n+1} = (11-1)^{3} = 10^{3} = 1000 \]The whole-number condition rules out other roots such as \( 121^{1} \) with \( n = 1 \), which would need \( m = 121 \) - and the standard convention takes the prime-power form.

Q: Find the value of \((0.008)^{1/3}\).

Correct answer: 0.2Write the decimal as a fraction to make the cube root obvious.\[0.008 = \frac{8}{1000} = \left(\frac{2}{10}\right)^3\]\[(0.008)^{1/3} = \frac{2}{10} = 0.2\]

Important Instructions

21.

Find the value of \(\left(\dfrac{81}{16}\right)^{-3/4}\).

A.8/27
B.27/8
C.16/81
D.4/9

Answer: B

Correct answer: 8/27

Recognise the base as a fourth power.

\[\frac{81}{16} = \left(\frac{3}{2}\right)^4\]

\[\left(\left(\frac{3}{2}\right)^4\right)^{-3/4} = \left(\frac{3}{2}\right)^{-3} = \left(\frac{2}{3}\right)^3 = \frac{8}{27}\]

The negative sign in the index inverts the fraction; the denominator 4 takes the fourth root and the numerator 3 cubes the result.

Discuss in Forum

22.

If \(m\) and \(n\) are whole numbers such that \(m^{n}=121\), the value of \((m-1)^{n+1}\) is:

A.1
B.10
C.121
D.1000

Answer: D

\( 121 = 11^{2} \), so \( m = 11 \) and \( n = 2 \).

\[ (m-1)^{n+1} = (11-1)^{3} = 10^{3} = 1000 \]

The whole-number condition rules out other roots such as \( 121^{1} \) with \( n = 1 \), which would need \( m = 121 \) - and the standard convention takes the prime-power form.

Discuss in Forum

23.

\(\frac{1}{1+x^{(b-a)}+x^{(c-a)}}+\frac{1}{1+x^{(a-b)}+x^{(c-b)}}+\frac{1}{1+x^{(b-c)}+x^{(a-c)}}=?\)

A.1
B.x^(a-b-c)
C.None of these

Answer: A

Given Exp.= \(\frac{1}{(1+\frac{x^{b}}{x^{a}}+\frac{x^{c}}{x^{a}})}+\frac{1}{(1+\frac{x^{a}}{x^{b}}+\frac{x^{c}}{x^{b}})}+\frac{1}{(1+\frac{x^{b}}{x^{c}}+\frac{x^{a}}{x^{c}})}\)

\(=\frac{x^{a}}{(x^{a}+x^{b}+x^{c})}+\frac{x^{b}}{(x^{a}+x^{b}+x^{c})}+\frac{x^{c}}{(x^{a}+x^{b}+x^{c})}\)

\(=\frac{(x^{a}+x^{b}+x^{c})}{(x^{a}+x^{b}+x^{c})} \)

\(=1\)