Question
Simplify the given expression. Write the answer with positive exponents. Assume that all variables represent positive numbers.
StartFraction left parenthesis y Superscript 5 Baseline z right parenthesis Superscript one tenth Over y Superscript negative one half Baseline z Superscript one fifth EndFraction
y5z
1
10
y−
1
2z
1
5
Question content area bottom
Part 1
StartFraction left parenthesis y Superscript 5 Baseline z right parenthesis Superscript one tenth Over y Superscript negative one half Baseline z Superscript one fifth EndFraction
y5z
1
10
y−
1
2z
1
5equals= enter your response here
StartFraction left parenthesis y Superscript 5 Baseline z right parenthesis Superscript one tenth Over y Superscript negative one half Baseline z Superscript one fifth EndFraction
y5z
1
10
y−
1
2z
1
5
Question content area bottom
Part 1
StartFraction left parenthesis y Superscript 5 Baseline z right parenthesis Superscript one tenth Over y Superscript negative one half Baseline z Superscript one fifth EndFraction
y5z
1
10
y−
1
2z
1
5equals= enter your response here
Answers
GPT-4o mini
To simplify the expression
\[
\frac{(y^5 z)^{\frac{1}{10}}}{y^{-\frac{1}{2}} z^{\frac{1}{5}}}
\]
we will first simplify the numerator and denominator separately.
### Step 1: Simplifying the numerator
The numerator is \((y^5 z)^{\frac{1}{10}}\).
Using the power of a product rule \((ab)^m = a^m b^m\):
\[
(y^5 z)^{\frac{1}{10}} = (y^5)^{\frac{1}{10}} (z)^{\frac{1}{10}} = y^{\frac{5}{10}} z^{\frac{1}{10}} = y^{\frac{1}{2}} z^{\frac{1}{10}}.
\]
### Step 2: Writing the denominator
The denominator is \(y^{-\frac{1}{2}} z^{\frac{1}{5}}\).
### Step 3: Rewrite the entire expression
Now we have:
\[
\frac{y^{\frac{1}{2}} z^{\frac{1}{10}}}{y^{-\frac{1}{2}} z^{\frac{1}{5}}}.
\]
### Step 4: Simplifying using properties of exponents
Applying \(a^m / a^n = a^{m-n}\):
For \(y\):
\[
y^{\frac{1}{2}} / y^{-\frac{1}{2}} = y^{\frac{1}{2} - (-\frac{1}{2})} = y^{\frac{1}{2} + \frac{1}{2}} = y^1 = y.
\]
For \(z\):
\[
z^{\frac{1}{10}} / z^{\frac{1}{5}} = z^{\frac{1}{10} - \frac{1}{5}} = z^{\frac{1}{10} - \frac{2}{10}} = z^{-\frac{1}{10}}.
\]
### Step 5: Final Expression
Putting it all together we have:
\[
y z^{-\frac{1}{10}}.
\]
### Step 6: Write with positive exponents
To express \(z^{-\frac{1}{10}}\) with a positive exponent, we can write it as:
\[
\frac{y}{z^{\frac{1}{10}}}.
\]
### Final Answer
Thus, the simplified expression is:
\[
\frac{y}{z^{\frac{1}{10}}}.
\]
\[
\frac{(y^5 z)^{\frac{1}{10}}}{y^{-\frac{1}{2}} z^{\frac{1}{5}}}
\]
we will first simplify the numerator and denominator separately.
### Step 1: Simplifying the numerator
The numerator is \((y^5 z)^{\frac{1}{10}}\).
Using the power of a product rule \((ab)^m = a^m b^m\):
\[
(y^5 z)^{\frac{1}{10}} = (y^5)^{\frac{1}{10}} (z)^{\frac{1}{10}} = y^{\frac{5}{10}} z^{\frac{1}{10}} = y^{\frac{1}{2}} z^{\frac{1}{10}}.
\]
### Step 2: Writing the denominator
The denominator is \(y^{-\frac{1}{2}} z^{\frac{1}{5}}\).
### Step 3: Rewrite the entire expression
Now we have:
\[
\frac{y^{\frac{1}{2}} z^{\frac{1}{10}}}{y^{-\frac{1}{2}} z^{\frac{1}{5}}}.
\]
### Step 4: Simplifying using properties of exponents
Applying \(a^m / a^n = a^{m-n}\):
For \(y\):
\[
y^{\frac{1}{2}} / y^{-\frac{1}{2}} = y^{\frac{1}{2} - (-\frac{1}{2})} = y^{\frac{1}{2} + \frac{1}{2}} = y^1 = y.
\]
For \(z\):
\[
z^{\frac{1}{10}} / z^{\frac{1}{5}} = z^{\frac{1}{10} - \frac{1}{5}} = z^{\frac{1}{10} - \frac{2}{10}} = z^{-\frac{1}{10}}.
\]
### Step 5: Final Expression
Putting it all together we have:
\[
y z^{-\frac{1}{10}}.
\]
### Step 6: Write with positive exponents
To express \(z^{-\frac{1}{10}}\) with a positive exponent, we can write it as:
\[
\frac{y}{z^{\frac{1}{10}}}.
\]
### Final Answer
Thus, the simplified expression is:
\[
\frac{y}{z^{\frac{1}{10}}}.
\]