To solve the expression and the equation given:
6. [tex]\(\frac{45 y^3 z^{10}}{5 y^3 z} y^{3-3} = 10 - 1\)[/tex]
Let's break it down step-by-step:
1. Simplify the fraction:
[tex]\[
\frac{45 y^3 z^{10}}{5 y^3 z}
\][/tex]
- Divide the coefficients: [tex]\(\frac{45}{5} = 9\)[/tex].
- Simplify the [tex]\(y\)[/tex] terms: [tex]\(y^3\)[/tex] in the numerator and denominator cancel each other.
- Simplify the [tex]\(z\)[/tex] terms: [tex]\(z^{10}\)[/tex] divided by [tex]\(z\)[/tex] is [tex]\(z^{10-1} = z^9\)[/tex].
Thus, the fraction simplifies to:
[tex]\[
9z^9
\][/tex]
2. Incorporate the [tex]\(y^{3-3}\)[/tex] term:
[tex]\[
9z^9 \cdot y^{3-3}
\][/tex]
- [tex]\(y^{3-3} = y^{0}\)[/tex], and we know that any term to the power of 0 is 1:
[tex]\[
y^0 = 1
\][/tex]
Therefore, multiplying by 1 does not change the value:
[tex]\[
9z^9 \cdot 1 = 9z^9
\][/tex]
3. Simplifying the right side:
[tex]\[
10 - 1
\][/tex]
- This is a simple arithmetic operation:
[tex]\[
10 - 1 = 9
\][/tex]
4. Combine and equate both sides:
[tex]\[
9z^9 = 9
\][/tex]
So, the expression simplifies to [tex]\(9z^9 = 9\)[/tex].
To summarize:
- The given fraction simplifies to [tex]\(9z^9\)[/tex].
- Incorporating the [tex]\(y^{3-3}\)[/tex] term results in [tex]\(9z^9\)[/tex].
- The right-hand side simplifies to [tex]\(9\)[/tex].
Therefore, the final result is:
[tex]\[
9z^9 - 1
\][/tex]
