Question

What values of a and b make the equation true? StartRoot 648 EndRoot = StartRoot 2 Superscript a Baseline times 3 Superscript b Baseline EndRoot a = 3, b = 2 a = 2, b = 3 a = 3, b = 4 a = 4, b = 3

Answer 1

Option C: $a=3, b=4$ is the value of a and b

Explanation:

Given that the expression $\sqrt{648}=\sqrt{2^a\times3^b}$

We need to determine the value of a and b

Let us consider the term $\sqrt{648}$ and take the prime factorization of the term 648

Thus, we have,

648 divides by 2,

$648=2 \cdot 324$

324 divides by 2,

$648=2 \cdot 2 \cdot 162$

162 divides by 2,

$648=2 \cdot 2 \cdot 2 \cdot 81$

81 divides by 3,

$648=2 \cdot 2 \cdot 2 \cdot 3 \cdot 27$

27 divides by 3,

$648=2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 9$

9 divides by 3,

$648=2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3$

Thus, we have,

$\sqrt{648}=\sqrt{2^{3} \cdot 3^{4}}$

Therefore, equating the powers of 2 and 3, we get,

$a=3, b=4$

Hence, the value of a and b is 3 and 4

Thus, Option C is the correct answer.

Answer 2

The values of a and b which make the equation true are; a = 3 and b = 4 respectively.

From the question; the expression given is;

$\sqrt{648} = \sqrt{ {2}^{a} \times {3}^{b} }$

In essence; we have;

• 648 = 2^a × 3^b

By expression of of 648 as the product of its lowest factors; we have;

• 648 = 2 × 2 × 2 × 3 × 3 × 3 × 3.

In which case; we have;

• 648 = 2³ × 3⁴

Ultimately, the values of a and b which make the equation true are 3 and 4 respectively.

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