256. Add the exponents when multiplying powers of the same base: 2^5 2^3 = 2^8 = 256.

8. A negative exponent inverts the base: ( 12 )^-3 = 2^3 = 8.

If 2^x+3 = 32, find x.

2. 32 = 2^5, so x + 3 = 5 and x = 2.

Simplify 5^n+2 - 6 5^n13 5^n - 2 5^n+1.

193. Factor 5^n from the top and bottom. Top: 5^n(5^2 - 6) = 19 5^n. Bottom: 5^n(13 - 2 5) = 3 5^n. The ratio is 193.

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Quantitative Aptitude · Numbers & Simplification

Surds & Indices — Questions & Answers

Surds and indices questions rest on the laws of exponents. Most reduce to writing both sides of an equation with the same base, or to comparing surds by raising them to a common power.

5 solved questions
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Step-by-step AI walkthroughs
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Key formulas included

Key formulas

$a^m \cdot a^n = a^{m+n}$ , $\;\dfrac{a^m}{a^n} = a^{m-n}$ , $\;(a^m)^n = a^{mn}$
$a^0 = 1$ , $\;a^{-n} = \dfrac{1}{a^n}$ , $\;a^{1/n} = \sqrt[n]{a}$
$(ab)^n = a^n b^n$ and $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$

5 solved Surds & Indices questions

Try each question first — tap an option to check your answer, or open Learn step by step for a guided walkthrough.

01
Evaluate $2^5 \times 2^3$ .
5 steps · reveal one at a time
1. Step 1: Identify the expression
  We need to evaluate the expression $2^5 \times 2^3$ . Notice that both terms have the same base, which is 2.
2. Step 2: Recall exponent rule
  When multiplying powers with the same base, you add their exponents. The rule is $a^m \times a^n = a^{m+n}$ .
3. Step 3: Apply the rule
  Using the rule, we add the exponents 5 and 3. So, $2^5 \times 2^3 = 2^{5+3} = 2^8$ .
4. Step 4: Calculate the value
  Now, we need to calculate $2^8$ . This means multiplying 2 by itself 8 times: $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 256$ .
5. Step 5: The answer
  So the answer is (B) 256. We applied the exponent rule for multiplication and then calculated the final power.
02
Evaluate $\left(\dfrac{1}{2}\right)^{-3}$ .
5 steps · reveal one at a time
1. Step 1: Identify the expression
  We need to evaluate the expression $\left(\dfrac{1}{2}\right)^{-3}$ . This involves a fraction as a base and a negative exponent.
2. Step 2: Recall negative exponents
  Remember that a negative exponent means you take the reciprocal of the base and make the exponent positive. Mathematically, $\left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^n$ .
3. Step 3: Apply the rule
  Applying this rule to our expression, we invert the base $\dfrac{1}{2}$ to get $2$ , and change the exponent from $-3$ to $3$ . So, $\left(\dfrac{1}{2}\right)^{-3} = 2^3$ .
4. Step 4: Calculate the power
  Now, we just need to calculate $2^3$ . This means multiplying $2$ by itself three times: $2 \times 2 \times 2 = 4 \times 2 = 8$ .
5. Step 5: The answer
  So the answer is (C) 8. This is the result we found after inverting the base and evaluating the positive power.
03
If $3^{x} = 81$ , find $x$ .
5 steps · reveal one at a time
1. Step 1: Understand the equation
  We are given the equation $3^x = 81$ . Our goal is to find the value of $x$ that makes this statement true. This means we need to determine what power of 3 results in 81.
2. Step 2: Express 81 as a power of 3
  To solve for $x$ , it's helpful to express 81 as a power of the base 3. Let's start by multiplying 3 by itself: $3^1 = 3$ , $3^2 = 9$ , $3^3 = 27$ .
3. Step 3: Calculate the exponent
  Continuing our multiplication from the previous step: $3^4 = 3 \times 3^3 = 3 \times 27 = 81$ . We have successfully expressed 81 as $3^4$ .
4. Step 4: Compare exponents
  Now we can substitute this back into our original equation: $3^x = 3^4$ . Since the bases are the same, the exponents must be equal. Therefore, $x = 4$ .
5. Step 5: The answer
  So the answer is (B) 4. We found that $3^4 = 81$ , which means $x$ must be 4.
04
If $2^{x+3} = 32$ , find $x$ .
5 steps · reveal one at a time
1. Step 1: Understand the Equation
  We are given the equation $2^{x+3} = 32$ . Our goal is to find the value of $x$ . To do this, we should try to express both sides of the equation with the same base.
2. Step 2: Convert Base
  Let's express the number 32 as a power of 2. We can calculate: $2^1=2$ , $2^2=4$ , $2^3=8$ , $2^4=16$ , and $2^5=32$ . So, we can rewrite the equation as $2^{x+3} = 2^5$ .
3. Step 3: Equate Exponents
  Since the bases on both sides of the equation are now the same (both are 2), their exponents must also be equal. This means we can set the exponents equal to each other: $x+3 = 5$ .
4. Step 4: Solve for x
  Now we have a simple linear equation to solve for $x$ . Subtract 3 from both sides of the equation: $x = 5 - 3$ . This gives us $x = 2$ .
5. Step 5: The answer
  So the answer is (D) 2. We found that $x=2$ by expressing 32 as $2^5$ and then equating the exponents.
05
Simplify $\dfrac{5^{n+2} - 6 \cdot 5^{n}}{13 \cdot 5^{n} - 2 \cdot 5^{n+1}}$ .
4 steps · reveal one at a time
1. Step 1: Simplify the Numerator
  First, let's simplify the numerator of the expression. We have $5^{n+2} - 6 \cdot 5^{n}$ . Remember that $5^{n+2}$ can be written as $5^n \cdot 5^2$ . This allows us to factor out $5^n$ , giving us $5^n(5^2 - 6) = 5^n(25 - 6) = 19 \cdot 5^n$ .
2. Step 2: Simplify the Denominator
  Next, we simplify the denominator: $13 \cdot 5^{n} - 2 \cdot 5^{n+1}$ . Similarly, $5^{n+1}$ can be written as $5^n \cdot 5^1$ . Factoring out $5^n$ from the denominator gives us $5^n(13 - 2 \cdot 5) = 5^n(13 - 10) = 3 \cdot 5^n$ .
3. Step 3: Combine and Solve
  Now we have the simplified numerator and denominator. The expression becomes $\dfrac{19 \cdot 5^n}{3 \cdot 5^n}$ . We can cancel out the common term $5^n$ from both the top and bottom. This leaves us with $\dfrac{19}{3}$ .
4. Step 4: The answer
  So the answer is (B) $\dfrac{19}{3}$ , as this is the simplified form of the given expression after factoring and canceling common terms.

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