Surds & Indices Questions and Answers

Answer: B

The value is 0.0333..., written \(0{\cdot}0\overline{3}\).

\[ \begin{aligned} x &= \sqrt{198 \times 550} = \sqrt{108900} = 330 \\ y &= \sqrt[3]{99 \times 363} = \sqrt[3]{35937} = 33 \\ \frac{1}{x} + \frac{1}{y} &= \frac{1}{330} + \frac{10}{330} \\ &= \frac{11}{330} = \frac{1}{30} \end{aligned} \]

1/30 = 0.0333..., where only the 3 repeats. 0.3333... is 1/3, ten times too large, 0.0101... is 1/99 and 0.111... is 1/9.

Answer: A

In standard form p is \(1{\cdot}1773 \times 10^{11}\).

Write both terms with the same power of ten before subtracting:

\[ \begin{aligned} p &= 123{\cdot}4 \times 10^{9} - 5{\cdot}67 \times 10^{9} \\ &= 117{\cdot}73 \times 10^{9} \\ &= 1{\cdot}1773 \times 10^{11} \end{aligned} \]

\(1{\cdot}1773 \times 10^{10}\) moves the decimal point one place too few. \(1{\cdot}773 \times 10^{11}\) and \(1{\cdot}773 \times 10^{10}\) drop a digit, as happens when the two numbers are subtracted without first matching their powers of ten.

Answer: A

The number is 7/16.

\[ \begin{aligned} \left(-\tfrac{1}{4}\right)^{-1} &= -4 \\ \left(-\tfrac{7}{4}\right)^{-1} &= -\tfrac{4}{7} \\ \text{its reciprocal} &= -\tfrac{7}{4} \\ -4 \times k &= -\tfrac{7}{4} \Rightarrow k = \tfrac{7}{16} \end{aligned} \]

A check: \(-4 \times \tfrac{7}{16} = -\tfrac{7}{4}\). The answer is positive because a negative number must be multiplied by a positive one to give another negative number, which rules out both negative options. 16/7 is the reciprocal of the correct answer.

Answer: C

The value of x is 3/5.

With the same base, multiplying powers means adding the exponents:

\[ \begin{aligned} \left(\tfrac{5}{7}\right)^{4} \times \left(\tfrac{5}{7}\right)^{-3} &= \left(\tfrac{5}{7}\right)^{1} \\ 5x - 2 &= 1 \\ x &= \tfrac{3}{5} \end{aligned} \]

1/5 comes from solving 5x − 2 = −1 (subtracting the exponents the wrong way), and the other values do not satisfy the equation.

Answer: B

The number is 10000.

Write 12500 in prime factors:

\[ 12500 = 125 \times 100 = 5^{5} \times 2^{2} \]

So \(x = 5\) and \(y = 2\). Then

\[ \begin{aligned} 5^{x - 1} \times 2^{2y} &= 5^{4} \times 2^{4} \\ &= 625 \times 16 = 10000 \end{aligned} \]

Equivalently, \(5^{4} \times 2^{4} = 10^{4}\). 2500 is \(5^{4} \times 2^{2}\), which forgets to double the power of 2, and 62500 is \(5^{6} \times 2^{2}\).

Answer: D

The value of \((2k + n)\) is 3.25.

Multiply the number parts and add the powers of ten, then write the result in standard form, with one non-zero digit before the decimal point.

\[ \begin{aligned} 1.7 \times 12.5 &= 21.25 \\ 10^{4} \times 10^{-6} &= 10^{-2} \\ 21.25 \times 10^{-2} &= 2.125 \times 10^{-1} \end{aligned} \]

So \(k = 2.125\) and \(n = -1\).

\[ 2k + n = 4.25 - 1 = 3.25 \]

2.125 is k alone, and 2.25 would come from taking \(n = -2\) without moving the decimal point into standard form.

Answer: D

The value is \(-\frac{48}{265}\).

A negative exponent inverts the fraction:

\[ \begin{aligned} \left(\tfrac{-1}{3}\right)^{-3} &= (-3)^{3} = -27 \\ \left(\tfrac{2}{3}\right)^{-4} &= \left(\tfrac{3}{2}\right)^{4} = \tfrac{81}{16} \\ \tfrac{x}{y} &= -27 \times \tfrac{16}{81} = -\tfrac{16}{3} \end{aligned} \]

\[ \begin{aligned} \tfrac{x}{y} + \tfrac{y}{x} &= -\tfrac{16}{3} - \tfrac{3}{16} = -\tfrac{265}{48} \\ \left(-\tfrac{265}{48}\right)^{-1} &= -\tfrac{48}{265} \end{aligned} \]

\(-\frac{3}{16}\) is only y/x, and the positive options miss the negative sign carried by the odd power of \(-\frac{1}{3}\).

Answer: B

The value of the square root is 5. First write the denominator in simple form: the square root of 48 is 4 times the square root of 3 and the square root of 12 is 2 times the square root of 3. Then multiply the top and bottom by the conjugate of the denominator.

\[ \frac{5 - 2\sqrt{3}}{7 + 4\sqrt{3}} \]

\[ \times \frac{7 - 4\sqrt{3}}{7 - 4\sqrt{3}} \]

\[ = \frac{35 - 20\sqrt{3} - 14\sqrt{3} + 24}{49 - 48} \]

\[ = 59 - 34\sqrt{3} \]

So a = 59 and b = -34, and a + b = 25. Then the square root of (a + b) is the square root of 25, which is 5. The options 4, 3 and 2 do not come from this value of a + b.

Answer: A

The value is 347.

The numerator is \(347^{0.5}\times 1388^{0.5}=\sqrt{347\times1388}=\sqrt{481636}=694\). The denominator is \(16^{0.16}\times 2^{0.36}=2^{0.64}\times2^{0.36}=2^{1}=2\), since 16 is 2 to the power 4.

\[ \frac{694}{2}=347 \]

694, 447 and 894 do not follow from these steps.

A quick check of the numerator: 347 × 1388 equals 481636, and 694 × 694 is also 481636, so the square root is exactly 694. The power 0.16 of 16 is 2 raised to 4 × 0.16 = 0.64, and 0.64 + 0.36 = 1, which makes the denominator 2.

Answer: A

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

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

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

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

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

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

\(=4^{1}\)

\(=4\)