Surds & Indices Questions and Answers

Q: Which among \(3\sqrt{8}\), 3, \(\sqrt[3]{1000}\), 5 is greatest ?

The greatest is \(\sqrt[3]{1000}\).Evaluate each number:\[ \sqrt[3]{1000} = 10, \quad \sqrt[3]{8} = 2 \]So the values are 10, 2, 3 and 5, and \(\sqrt[3]{1000} = 10\) is the greatest. The stem prints the first number as \(3\sqrt{8}\), while option C has \(\sqrt[3]{8}\); even read as \(3\sqrt{8} \approx 8.49\), it is still less than 10, so the answer does not change.

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.

Which among \(3\sqrt{8}\), 3, \(\sqrt[3]{1000}\), 5 is greatest ?

A.5
B.3
C.\(\sqrt[3]{8}\)
D.\(\sqrt[3]{1000}\)

Exam Year(s)

Tripura Public Service Commission - 2019,

Answer: D

The greatest is \(\sqrt[3]{1000}\).

Evaluate each number:

\[ \sqrt[3]{1000} = 10, \quad \sqrt[3]{8} = 2 \]

So the values are 10, 2, 3 and 5, and \(\sqrt[3]{1000} = 10\) is the greatest. The stem prints the first number as \(3\sqrt{8}\), while option C has \(\sqrt[3]{8}\); even read as \(3\sqrt{8} \approx 8.49\), it is still less than 10, so the answer does not change.

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22.

\((256)^{0.16}\times (256)^{0.09}=?\)

A.4
B.16
C.64
D.256.25

Answer: A

\((256)^{0.16}\times (256)^{0.09}=(256)^{(0.16+0.09)}\)

\(=(256)^{0.25}\)

\(=(256)^{(25/100)}\)

\(=(256)^{(1/4)}\)

\(=(4^{4})^{(1/4)}\)

\(=4^{4(1/4)}\)

\(=4^{1}\)

\(=4\)

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23.

\[ \frac{1}{1+a^{(n-m)}} + \frac{1}{1+a^{(m-n)}} = \ ? \]

A.0
B.1/2
C.1
D.\( a^{(m+n)} \)

Answer: C

\(\frac{1}{1+a^{(n-m)}}+\frac{1}{1+a^{(m-n)}} \)

\(= \frac{1}{(1+\frac{an}{am})}+\frac{1}{(1+\frac{am}{an})} \)

\(= \frac{am}{(am+an)}+\frac{an}{(am+an)} \)

\(= \frac{(am+an)}{(am+an)} \)

\(= 1\)

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24.

If \(2^x \times 16^{\frac{2}{5}} = 2^{\frac{1}{5}}\), then \(x\) is equal to:

A.2/5
B.-2/5
C.7/5
D.-7/5

Answer: D

\(2^x \times 16^{\frac{2}{5}} = 2^{\frac{1}{5}}\)

\(\Rightarrow 2^x \times (2^4)^{\frac{2}{5}} = 2^{\frac{1}{5}} \)

\(\Rightarrow 2^x \times 2^{\frac{8}{5}}= 2^{\frac{1}{5}} \)

\(\Rightarrow 2^{(x+ \frac{8}{5})} = 2^{\frac{1}{5}} \)

\(\Rightarrow x + \frac{8}{5} = \frac{1}{5} \)

\(\Rightarrow x = (\frac{1}{5} - \frac{8}{5}) = -\frac{7}{5}\)

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25.

\((25)^{7.5} \times (5)^{2.5} \div (125)^{1.5} = 5?\)

A.8.5
B.13
C.16
D.17.5

Answer: B

Let \((25)^{7.5} \times (5)^{2.5} \div (125)^{1.5} = 5x.\)

Then, \(\frac{(5)^{7.5} \times (5)^{2.5}}{(5^{3})^{1.5}} = 5x \)

\(\Rightarrow \frac{5^{(2 \times 7.5)} \times (5)^{2.5}}{5^{(3 \times 1.5)}} = 5x\)

\(\Rightarrow \frac{5^{15} \times 5^{2.5}}{5^{4.5}} = 5x\)

\(\Rightarrow 5x = 5^{(15+2.5-4.5)} \)

\(\Rightarrow 5x = 5^{13}, \therefore x = 13\)

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26.

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.

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27.

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.

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28.

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

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

Answer: A

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

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

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

\(=1\)

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29.

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

A.0.02
B.0.2
C.0.5
D.2

Answer: B

Correct answer: 0.2

Write the decimal as a fraction to make the cube root obvious.

\[0.008 = \frac{8}{1000} = \left(\frac{2}{10}\right)^3\]

\[(0.008)^{1/3} = \frac{2}{10} = 0.2\]

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30.

If \(\sqrt{(3+3\sqrt{x})} = 2\), then \(x\) is equal to:

Answer: A

Exp: On squaring both sides, we get:

\(3+3\sqrt{x} = 4\) or \(3\sqrt{x} = 1\)

Cubing both sides, we get \(x = (1\times 1\times 1) = 1\)

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