Answer :
To solve these problems, we need to understand the properties of exponents. Specifically, we use the property [tex]a^m \cdot a^n = a^{m+n}[/tex] for multiplication and [tex]\frac{a^m}{a^n} = a^{m-n}[/tex] for division.
Let's solve each problem:
- [tex]y^3 \cdot b^4 \cdot y^6 = y^9 b^n[/tex]
Combine the terms involving [tex]y[/tex]:
[tex]y^3 \cdot y^6 = y^{3+6} = y^9[/tex].
So the equation simplifies to: [tex]y^9 \cdot b^4 = y^9 b^n[/tex].
This shows [tex]b^4 = b^n[/tex], hence [tex]n = 4[/tex].
- [tex]a^3 \cdot c^4 \cdot c^3 \cdot a^7 = a^{10} c^n[/tex]
Combine the terms involving [tex]a[/tex]:
[tex]a^3 \cdot a^7 = a^{3+7} = a^{10}[/tex].
Combine the terms involving [tex]c[/tex]:
[tex]c^4 \cdot c^3 = c^{4+3} = c^7[/tex].
The equation becomes [tex]a^{10} c^7 = a^{10} c^n[/tex].
So [tex]n = 7[/tex].
- [tex]\frac{e^{14}}{e^n} = e^{10}[/tex]
Use the division property:
[tex]e^{14-n} = e^{10}[/tex].
Therefore, [tex]14 - n = 10[/tex].
Solving for [tex]n[/tex], [tex]n = 4[/tex].
- [tex]t \cdot v^6 \cdot t^2 = t^n v^6[/tex]
Combine the terms involving [tex]t[/tex]:
[tex]t^1 \cdot t^2 = t^{1+2} = t^3[/tex].
The equation becomes [tex]t^3 v^6 = t^n v^6[/tex].
Therefore, [tex]n = 3[/tex].
- [tex]i^n \div i^{10} = i^{35}[/tex]
Use the division property:
[tex]i^{n-10} = i^{35}[/tex].
Therefore, [tex]n - 10 = 35[/tex].
Solving for [tex]n[/tex], [tex]n = 45[/tex].
- [tex]\frac{w^{12}}{w^6} = w^n[/tex]
Use the division property:
[tex]w^{12-6} = w^n[/tex].
Therefore, [tex]n = 6[/tex].
- [tex]x^4 \cdot y^3 \cdot x^5 \cdot y^2 = x^n y^5[/tex]
Combine the terms involving [tex]x[/tex]:
[tex]x^4 \cdot x^5 = x^{4+5} = x^9[/tex].
Combine the terms involving [tex]y[/tex]:
[tex]y^3 \cdot y^2 = y^{3+2} = y^5[/tex].
The equation becomes [tex]x^9 y^5 = x^n y^5[/tex].
Therefore, [tex]n = 9[/tex].
- [tex]p^3 \cdot q^4 \cdot q^7 = p^n q^{11}[/tex]
Leave the [tex]p[/tex] term as it is:
[tex]p^3[/tex].
Combine the terms involving [tex]q[/tex]:
[tex]q^{4+7} = q^{11}[/tex].
The equation becomes [tex]p^3 q^{11} = p^n q^{11}[/tex].
Therefore, [tex]n = 3[/tex].
- [tex]z^n \div z^8 = z^6[/tex]
Use the division property:
[tex]z^{n-8} = z^6[/tex].
Therefore, [tex]n - 8 = 6[/tex].
Solving for [tex]n[/tex], [tex]n = 14[/tex].
- [tex]\frac{s^{20}}{s^n} = s^5[/tex]
Use the division property:
[tex]s^{20-n} = s^5[/tex].
Therefore, [tex]20 - n = 5[/tex].
Solving for [tex]n[/tex], [tex]n = 15[/tex].
