Question
\( (b)\left(5 a^{2} b^{3}\right)^{3} \times\left(x^{4}\right)^{-2} \) (i) \( 4^{2} \times 8^{2+1} \)
Ask by Evans Harper. in South Africa
Feb 25,2025
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Answer
1. \( b \times (5a^{2}b^{3})^{3} \times (x^{4})^{-2} = \frac{125a^{6}b^{10}}{x^{8}} \)
2. \( 4^{2} \times 8^{2+1} = 8192 \)
Solution
We are given two expressions to simplify.
1. Expression (b):
b · (5a²b³)³ × (x⁴)⁻²
Step 1: Expand (5a²b³)³. When raising a product to a power, raise each factor to that power:
(5)³ = 125
(a²)³ = a^(2×3) = a⁶
(b³)³ = b^(3×3) = b⁹
So, (5a²b³)³ = 125a⁶b⁹.
Step 2: Now multiply by the initial b:
b × 125a⁶b⁹ = 125a⁶b^(1+9) = 125a⁶b¹⁰.
Step 3: Simplify (x⁴)⁻² by using the power rule:
(x⁴)⁻² = x^(4×(–2)) = x^(–8).
Thus, the full expression becomes:
125a⁶b¹⁰ × x^(–8)
or equivalently, 125a⁶b¹⁰/x⁸.
2. Expression (i):
4² × 8^(2+1)
Step 1: First, compute the exponents:
4² = 16
2+1 = 3, so 8³ = 8×8×8 = 512
Step 2: Now multiply:
16 × 512 = 8192.
Alternatively, noticing that 4 = 2² and 8 = 2³, we have:
4² = (2²)² = 2⁴ and 8³ = (2³)³ = 2⁹, so:
2⁴ × 2⁹ = 2^(4+9) = 2¹³ = 8192.
Final Answers:
1. b(5a²b³)³(x⁴)⁻² = 125a⁶b¹⁰/x⁸
2. 4² × 8^(2+1) = 8192
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Simplify this solution Extra Insights
When you simplify \( (b)(5 a^{2} b^{3})^{3} \times (x^{4})^{-2} \), you’ll be playing with the power rules of exponents. Start by expanding the first part: \( (5 a^{2} b^{3})^3 = 125 a^{6} b^{9} \). Then multiply that by \( b \), which gives \( 125 a^6 b^{10} \). For the second part, \( (x^4)^{-2} \) transforms to \( \frac{1}{x^8} \) due to the negative exponent. Put it all together: \( \frac{125 a^6 b^{10}}{x^8} \). Voilà! Now, moving on to \( 4^2 \times 8^{2+1} \): First, calculate \( 4^2 = 16 \) and \( 8^{2+1} = 8^3 = 512 \). Now multiply them: \( 16 \times 512 = 8192 \). There you have it - a mathematical feast!
