To determine what number you should multiply by [tex]\( 2^3 \times 7^3 \)[/tex] to get a perfect square, let's break down the problem step-by-step.
1. Identify the given expression: The expression given is [tex]\( 2^3 \times 7^3 \)[/tex].
2. Understand perfect squares: A perfect square is a number that can be expressed as [tex]\( n^2 \)[/tex] for some integer [tex]\( n \)[/tex]. This means that in the prime factorization of a perfect square, every exponent must be even.
3. Analyze the exponents: In the expression [tex]\( 2^3 \times 7^3 \)[/tex], the exponents of the prime factors are 3 and 3 for the bases 2 and 7, respectively.
4. Make the exponents even:
- The exponent of 2 in [tex]\( 2^3 \)[/tex] is 3, which is odd. To make it even, we need an additional factor of 2.
- The exponent of 7 in [tex]\( 7^3 \)[/tex] is 3, which is also odd. To make it even, we need an additional factor of 7.
5. Determine the necessary multiplier:
- To make the exponent of 2 even, multiply by [tex]\( 2^1 = 2 \)[/tex].
- To make the exponent of 7 even, multiply by [tex]\( 7^1 = 7 \)[/tex].
6. Combine the factors: The number needed to make the exponents even is [tex]\( 2 \times 7 = 14 \)[/tex].
Therefore, the number you should multiply by [tex]\( 2^3 \times 7^3 \)[/tex] to make it a perfect square is 14.
So, the correct answer is:
(d) 14
