Answer :
To determine the appropriate law for simplifying the expression \(\left(\frac{p}{q}\right)^3\), let's carefully consider the options given:
1. Power of a Power: This law is used when you have an exponent raised to another exponent, such as \((a^m)^n = a^{m \cdot n}\). This does not apply to our expression because we are not dealing with nested exponents.
2. Power of a Quotient: This law states that when you take a fraction or quotient to a power, you can apply the exponent to both the numerator and the denominator individually. In mathematical terms, \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\). This matches our case where we have \(\left(\frac{p}{q}\right)^3\), meaning we would apply the exponent 3 to both \(p\) and \(q\), resulting in \(\frac{p^3}{q^3}\).
3. Quotient of Powers: This law is used when you are dividing two expressions with the same base and subtracting their exponents, such as \(\frac{a^m}{a^n} = a^{m-n}\). This does not apply here because our expression involves a power of a fraction, not the division of powers.
4. Power of a Product: This law states that if you are raising a product to a power, you can apply the power to each factor within the product individually, such as \((a \cdot b)^n = a^n \cdot b^n\). This does not apply either, as we are not dealing with a product.
Based on these explanations, the correct law to use in simplifying the expression \(\left(\frac{p}{q}\right)^3\) is:
Power of a Quotient.
1. Power of a Power: This law is used when you have an exponent raised to another exponent, such as \((a^m)^n = a^{m \cdot n}\). This does not apply to our expression because we are not dealing with nested exponents.
2. Power of a Quotient: This law states that when you take a fraction or quotient to a power, you can apply the exponent to both the numerator and the denominator individually. In mathematical terms, \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\). This matches our case where we have \(\left(\frac{p}{q}\right)^3\), meaning we would apply the exponent 3 to both \(p\) and \(q\), resulting in \(\frac{p^3}{q^3}\).
3. Quotient of Powers: This law is used when you are dividing two expressions with the same base and subtracting their exponents, such as \(\frac{a^m}{a^n} = a^{m-n}\). This does not apply here because our expression involves a power of a fraction, not the division of powers.
4. Power of a Product: This law states that if you are raising a product to a power, you can apply the power to each factor within the product individually, such as \((a \cdot b)^n = a^n \cdot b^n\). This does not apply either, as we are not dealing with a product.
Based on these explanations, the correct law to use in simplifying the expression \(\left(\frac{p}{q}\right)^3\) is:
Power of a Quotient.
