Answer :
Certainly! Let's tackle each problem step-by-step.
### Problem 13:
Question: By what number should \((4)^{-3}\) be multiplied so that the product becomes \(\frac{1}{16}\)?
Step-by-step solution:
1. Identify the expression and the target product:
- Expression: \((4)^{-3}\)
- Target product: \(\frac{1}{16}\)
2. Rewrite the equation: We need to find a number, let's call it \(x\), that satisfies:
[tex]\[ (4)^{-3} \times x = \frac{1}{16} \][/tex]
3. Calculate \((4)^{-3}\):
[tex]\[ (4)^{-3} = \frac{1}{(4)^3} = \frac{1}{64} \][/tex]
4. Set up the equation with the target product:
[tex]\[ \frac{1}{64} \times x = \frac{1}{16} \][/tex]
5. Solve for \(x\):
[tex]\[ x = \frac{1}{16} \div \frac{1}{64} = \frac{1}{16} \times 64 = 4 \][/tex]
Conclusion:
The number that \((4)^{-3}\) should be multiplied by to get \(\frac{1}{16}\) is \(4\).
### Problem 14:
Question: By what number should \(\left(\frac{3}{5}\right)^{-2}\) be divided so that the quotient becomes 25?
Step-by-step solution:
1. Identify the expression and the target quotient:
- Expression: \(\left(\frac{3}{5}\right)^{-2}\)
- Target quotient: 25
2. Rewrite the equation: We need to find a number, let's call it \(y\), that satisfies:
[tex]\[ \frac{\left( \frac{3}{5} \right)^{-2}}{y} = 25 \][/tex]
3. Calculate \(\left(\frac{3}{5}\right)^{-2}\):
[tex]\[ \left(\frac{3}{5}\right)^{-2} = \left(\frac{5}{3}\right)^2 = \frac{25}{9} \][/tex]
4. Set up the equation with the target quotient:
[tex]\[ \frac{\frac{25}{9}}{y} = 25 \][/tex]
5. Solve for \(y\):
[tex]\[ y = \frac{25}{9} \div 25 = \frac{25}{9} \times \frac{1}{25} = \frac{1}{9} \][/tex]
Conclusion:
The number that [tex]\(\left(\frac{3}{5}\right)^{-2}\)[/tex] should be divided by to get 25 is [tex]\(\frac{1}{9}\)[/tex].
### Problem 13:
Question: By what number should \((4)^{-3}\) be multiplied so that the product becomes \(\frac{1}{16}\)?
Step-by-step solution:
1. Identify the expression and the target product:
- Expression: \((4)^{-3}\)
- Target product: \(\frac{1}{16}\)
2. Rewrite the equation: We need to find a number, let's call it \(x\), that satisfies:
[tex]\[ (4)^{-3} \times x = \frac{1}{16} \][/tex]
3. Calculate \((4)^{-3}\):
[tex]\[ (4)^{-3} = \frac{1}{(4)^3} = \frac{1}{64} \][/tex]
4. Set up the equation with the target product:
[tex]\[ \frac{1}{64} \times x = \frac{1}{16} \][/tex]
5. Solve for \(x\):
[tex]\[ x = \frac{1}{16} \div \frac{1}{64} = \frac{1}{16} \times 64 = 4 \][/tex]
Conclusion:
The number that \((4)^{-3}\) should be multiplied by to get \(\frac{1}{16}\) is \(4\).
### Problem 14:
Question: By what number should \(\left(\frac{3}{5}\right)^{-2}\) be divided so that the quotient becomes 25?
Step-by-step solution:
1. Identify the expression and the target quotient:
- Expression: \(\left(\frac{3}{5}\right)^{-2}\)
- Target quotient: 25
2. Rewrite the equation: We need to find a number, let's call it \(y\), that satisfies:
[tex]\[ \frac{\left( \frac{3}{5} \right)^{-2}}{y} = 25 \][/tex]
3. Calculate \(\left(\frac{3}{5}\right)^{-2}\):
[tex]\[ \left(\frac{3}{5}\right)^{-2} = \left(\frac{5}{3}\right)^2 = \frac{25}{9} \][/tex]
4. Set up the equation with the target quotient:
[tex]\[ \frac{\frac{25}{9}}{y} = 25 \][/tex]
5. Solve for \(y\):
[tex]\[ y = \frac{25}{9} \div 25 = \frac{25}{9} \times \frac{1}{25} = \frac{1}{9} \][/tex]
Conclusion:
The number that [tex]\(\left(\frac{3}{5}\right)^{-2}\)[/tex] should be divided by to get 25 is [tex]\(\frac{1}{9}\)[/tex].
