Question

evaluate. simplify your answer. \sqrt{\dfrac{4 + 2^2 \times 3}{9}}

Explanation:

Step1: Calculate exponent and multiplication

First, calculate \(2^2 = 4\), then multiply by 3: \(4\times3 = 12\).

Step2: Add in the numerator

Add 4 to the result: \(4 + 12 = 16\).

Step3: Divide by denominator

Divide the numerator by 9? Wait, no, wait the expression is \(\sqrt{\frac{4 + 2^2\times3}{9}}\)? Wait, looking at the image, the expression is \(\sqrt{\frac{4 + 2^2\times3}{9}}\)? Wait, no, let's re - examine. The numerator is \(4+2^{2}\times3\), denominator is 9. Wait, no, maybe I misread. Wait, the expression is \(\sqrt{\frac{4 + 2^{2}\times3}{9}}\)? Wait, no, let's do the order of operations. Inside the square root, numerator: \(4+2^{2}\times3\). First, exponent: \(2^{2}=4\), then multiplication: \(4\times3 = 12\), then addition: \(4 + 12=16\). Then the fraction is \(\frac{16}{9}\), and the square root of \(\frac{16}{9}\) is \(\frac{4}{3}\) because \(\sqrt{\frac{16}{9}}=\frac{\sqrt{16}}{\sqrt{9}}=\frac{4}{3}\). Wait, maybe I made a mistake in the denominator. Wait, the original expression: let's look again. The image shows \(\sqrt{\frac{4 + 2^{2}\times3}{9}}\)? Wait, no, maybe the denominator is 9? Wait, no, maybe the expression is \(\sqrt{4 + 2^{2}\times3}\div9\)? No, the layout: the square root covers \(\frac{4 + 2^{2}\times3}{9}\). So:

First, compute the numerator: \(4+2^{2}\times3\).

• Calculate \(2^{2}=4\)
• Then \(4\times3 = 12\)
• Then \(4 + 12=16\)

Then the fraction is \(\frac{16}{9}\), and the square root of \(\frac{16}{9}\) is \(\frac{4}{3}\) (since \(\sqrt{16}=4\) and \(\sqrt{9}=3\)).

Answer:

\(\frac{4}{3}\)