Answer :
To find the missing term in the denominator, we need to simplify both sides of the equation step by step.
1. Simplify \(\frac{7^{16}}{7^{12}}\) using the properties of exponents. According to the properties of exponents, \(\frac{a^m}{a^n} = a^{m-n}\). Thus,
[tex]\[ \frac{7^{16}}{7^{12}} = 7^{16-12} = 7^{4} \][/tex]
So, the left side of the equation simplifies to \(7^4\).
2. The equation now looks like this:
[tex]\[ 7^4 = \frac{7^{-18}}{?} \][/tex]
3. To isolate the missing term in the denominator, let's introduce a variable for it. Let’s call the missing term \(x\). Now the equation becomes:
[tex]\[ 7^4 = \frac{7^{-18}}{x} \][/tex]
4. We need to solve for \(x\). We know that \(\frac{7^{-18}}{x} = 7^4\). This means that
[tex]\[ 7^4 \cdot x = 7^{-18} \][/tex]
5. To balance the equation and solve for \(x\), we need to equate the exponents of 7 on both sides. Let’s rewrite \(x\) in exponential form as \(7^k\). Then,
[tex]\[ 7^4 \cdot 7^k = 7^{-18} \][/tex]
6. Using exponent addition properties \(a^m \cdot a^n = a^{m+n}\), this equation becomes:
[tex]\[ 7^{4+k} = 7^{-18} \][/tex]
7. Now, equate the exponents:
[tex]\[ 4 + k = -18 \][/tex]
8. Solve for \(k\):
[tex]\[ k = -18 - 4 = -22 \][/tex]
Therefore, the missing term in the denominator is \(7^{-22}\).
So, the correct answer from the drop-down menu is:
[tex]\[ \boxed{7^{-22}} \][/tex]
1. Simplify \(\frac{7^{16}}{7^{12}}\) using the properties of exponents. According to the properties of exponents, \(\frac{a^m}{a^n} = a^{m-n}\). Thus,
[tex]\[ \frac{7^{16}}{7^{12}} = 7^{16-12} = 7^{4} \][/tex]
So, the left side of the equation simplifies to \(7^4\).
2. The equation now looks like this:
[tex]\[ 7^4 = \frac{7^{-18}}{?} \][/tex]
3. To isolate the missing term in the denominator, let's introduce a variable for it. Let’s call the missing term \(x\). Now the equation becomes:
[tex]\[ 7^4 = \frac{7^{-18}}{x} \][/tex]
4. We need to solve for \(x\). We know that \(\frac{7^{-18}}{x} = 7^4\). This means that
[tex]\[ 7^4 \cdot x = 7^{-18} \][/tex]
5. To balance the equation and solve for \(x\), we need to equate the exponents of 7 on both sides. Let’s rewrite \(x\) in exponential form as \(7^k\). Then,
[tex]\[ 7^4 \cdot 7^k = 7^{-18} \][/tex]
6. Using exponent addition properties \(a^m \cdot a^n = a^{m+n}\), this equation becomes:
[tex]\[ 7^{4+k} = 7^{-18} \][/tex]
7. Now, equate the exponents:
[tex]\[ 4 + k = -18 \][/tex]
8. Solve for \(k\):
[tex]\[ k = -18 - 4 = -22 \][/tex]
Therefore, the missing term in the denominator is \(7^{-22}\).
So, the correct answer from the drop-down menu is:
[tex]\[ \boxed{7^{-22}} \][/tex]
