Answer :
To determine which equation shows that you can find the power of a power by finding the product of the exponents, let's explore the process with the given options one by one.
Let's consider the expression \((y^a)^b\). The rule for finding the power of a power is to multiply the exponents. This means we should have the form \(y\) with an exponent that is the product of \(a\) and \(b\).
Now, examining each option:
1. \(\left(y^a\right)^b = y^a \cdot y^b\)
- This equation implies that \((y^a)^b\) equals the product of \(y^a\) and \(y^b\). This is not consistent with our rule because it does not result in a single exponent that is the product of \(a\) and \(b\).
2. \(\left(y^a\right)^b = y^{a+b}\)
- This equation states that \((y^a)^b\) equals \(y\) raised to the power \(a + b\). Adding the exponents is incorrect because it does not follow the power of a power rule which requires multiplication of exponents.
3. \(\left(y^a\right)^b = y^a + y^b\)
- This interprets \((y^a)^b\) as the sum of two separate terms \(y^a\) and \(y^b\). This is irrelevant because we must keep the terms within a single base \(y\), with a single exponent.
4. \(\left(y^a\right)^b = y^{a \cdot b}\)
- This is in line with our rule: you multiply \(a\) and \(b\) to get the new exponent. Therefore, \(
\left(y^a\right)^b = y^{a \cdot b}
\) accurately represents the power of a power rule.
Hence, the correct equation is:
\(\left(y^a\right)^b = y^{a \cdot b}\)
So the correct option is the fourth one.
Let's consider the expression \((y^a)^b\). The rule for finding the power of a power is to multiply the exponents. This means we should have the form \(y\) with an exponent that is the product of \(a\) and \(b\).
Now, examining each option:
1. \(\left(y^a\right)^b = y^a \cdot y^b\)
- This equation implies that \((y^a)^b\) equals the product of \(y^a\) and \(y^b\). This is not consistent with our rule because it does not result in a single exponent that is the product of \(a\) and \(b\).
2. \(\left(y^a\right)^b = y^{a+b}\)
- This equation states that \((y^a)^b\) equals \(y\) raised to the power \(a + b\). Adding the exponents is incorrect because it does not follow the power of a power rule which requires multiplication of exponents.
3. \(\left(y^a\right)^b = y^a + y^b\)
- This interprets \((y^a)^b\) as the sum of two separate terms \(y^a\) and \(y^b\). This is irrelevant because we must keep the terms within a single base \(y\), with a single exponent.
4. \(\left(y^a\right)^b = y^{a \cdot b}\)
- This is in line with our rule: you multiply \(a\) and \(b\) to get the new exponent. Therefore, \(
\left(y^a\right)^b = y^{a \cdot b}
\) accurately represents the power of a power rule.
Hence, the correct equation is:
\(\left(y^a\right)^b = y^{a \cdot b}\)
So the correct option is the fourth one.
