Question

Simplify. $$\text { a. }\left(\frac{2}{9}\right)^{2}$$ $$\text { b. }\left(-\frac{3}{5}\right)^{3}$$ $$C_{*}\left(-\frac{4}{7}\right)^{2}$$ $$\text { d. }\left(\frac{2}{3}\right)^{5}$$ $$\text { e. } \sqrt{\frac{36}{81}}$$ $$\text { f. } \sqrt{\frac{64}{100}}$$ $$\text { g. } \sqrt{\frac{144}{49}}$$ $$\text { h. } \sqrt{\frac{375}{15}}$$

Short answer

Question: Simplify the given expression: \(\sqrt{\frac{64}{100}}\). Answer: \(\frac{4}{5}\).

Step-by-step solution

(Expression a)

For the expression \(\left(\frac{2}{9}\right)^{2}\), apply the exponent to both the numerator and the denominator. $$\left(\frac{2}{9}\right)^{2} = \frac{2^2}{9^2} = \frac{4}{81}$$

(Expression b)

For the expression \(\left(-\frac{3}{5}\right)^{3}\), apply the exponent to both the numerator and the denominator, and the negative sign to the numerator. $$\left(-\frac{3}{5}\right)^{3} = \frac{(-3)^3}{5^3} = \frac{-27}{125}$$

(Expression c)

For the expression \(\left(-\frac{4}{7}\right)^{2}\), remember that squaring a negative number yields a positive result. Apply the exponent to both the numerator and the denominator. $$\left(-\frac{4}{7}\right)^{2} = \frac{(-4)^2}{7^2} = \frac{16}{49}$$

(Expression d)

For the expression \(\left(\frac{2}{3}\right)^{5}\), apply the exponent to both the numerator and the denominator. $$\left(\frac{2}{3}\right)^{5} = \frac{2^5}{3^5} = \frac{32}{243}$$

(Expression e)

For the expression \(\sqrt{\frac{36}{81}}\), apply the square root to both the numerator and the denominator. $$\sqrt{\frac{36}{81}} = \frac{\sqrt{36}}{\sqrt{81}} = \frac{6}{9} = \frac{2}{3}$$

(Expression f)

For the expression \(\sqrt{\frac{64}{100}}\), apply the square root to both the numerator and the denominator. $$\sqrt{\frac{64}{100}} = \frac{\sqrt{64}}{\sqrt{100}} = \frac{8}{10} = \frac{4}{5}$$

(Expression g)

For the expression \(\sqrt{\frac{144}{49}}\), apply the square root to both the numerator and the denominator. $$\sqrt{\frac{144}{49}} = \frac{\sqrt{144}}{\sqrt{49}} = \frac{12}{7}$$

(Expression h)

For the expression \(\sqrt{\frac{375}{15}}\), first simplify the fraction, then apply the square root. $$\sqrt{\frac{375}{15}} = \sqrt{25} = 5$$