8 $sqrt{\frac{4a^{16}}{a^{12}}}$ при $a = 5$ 16 $sqrt{\frac{9x^{4}}{y^{6}}}$ при $x = 9$, $y = 3$
8 $sqrt{\frac{4a^{16}}{a^{12}}}$ при $a = 5$ 16 $sqrt{\frac{9x^{4}}{y^{6}}}$ при $x = 9$, $y = 3$
Answer
- For the first - expression (\sqrt{\frac{4a^{16}}{a^{12}}}) when (a = 5):
- Step 1: Simplify the expression inside the square - root using the rule of exponents (\frac{x^m}{x^n}=x^{m - n})
- (\frac{4a^{16}}{a^{12}}=4a^{16-12}=4a^{4}). So the original expression becomes (\sqrt{4a^{4}}).
- Step 2: Use the property (\sqrt{xy}=\sqrt{x}\cdot\sqrt{y}) ((x = 4), (y = a^{4}))
- (\sqrt{4a^{4}}=\sqrt{4}\cdot\sqrt{a^{4}}). Since (\sqrt{4}=2) and (\sqrt{a^{4}}=a^{2}), the simplified expression is (2a^{2}).
- Step 3: Substitute (a = 5) into the expression (2a^{2})
- When (a = 5), (2a^{2}=2\times5^{2}=2\times25 = 50).
- Step 1: Simplify the expression inside the square - root using the rule of exponents (\frac{x^m}{x^n}=x^{m - n})
- For the second - expression (\sqrt{\frac{9x^{4}}{y^{6}}}) when (x = 9) and (y = 3):
- Step 1: Simplify the expression inside the square - root using the rule (\sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}}) and (\sqrt{xy}=\sqrt{x}\cdot\sqrt{y})
- (\sqrt{\frac{9x^{4}}{y^{6}}}=\frac{\sqrt{9x^{4}}}{\sqrt{y^{6}}}). Since (\sqrt{9x^{4}}=\sqrt{9}\cdot\sqrt{x^{4}} = 3x^{2}) and (\sqrt{y^{6}}=y^{3}), the simplified expression is (\frac{3x^{2}}{y^{3}}).
- Step 2: Substitute (x = 9) and (y = 3) into the expression (\frac{3x^{2}}{y^{3}})
- When (x = 9) and (y = 3), we have (\frac{3\times9^{2}}{3^{3}}). First, (9^{2}=81), so the numerator is (3\times81 = 243), and the denominator is (3^{3}=27). Then (\frac{243}{27}=9).
- Step 1: Simplify the expression inside the square - root using the rule (\sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}}) and (\sqrt{xy}=\sqrt{x}\cdot\sqrt{y})
Answer:
The value of (\sqrt{\frac{4a^{16}}{a^{12}}}) when (a = 5) is (50), and the value of (\sqrt{\frac{9x^{4}}{y^{6}}}) when (x = 9) and (y = 3) is (9).