Let's solve the given expression step-by-step:
We start with the given expression:
[tex]\[ \sqrt[5]{10^3} \cdot \sqrt[5]{10^4} \][/tex]
First, we recognize that this expression involves roots and can be rewritten using exponents. Specifically, the fifth root of a number can be expressed as raising that number to the power of [tex]\( \frac{1}{5} \)[/tex]. Thus:
[tex]\[ \sqrt[5]{10^3} = (10^3)^{\frac{1}{5}} = 10^{3 \cdot \frac{1}{5}} = 10^{\frac{3}{5}} \][/tex]
[tex]\[ \sqrt[5]{10^4} = (10^4)^{\frac{1}{5}} = 10^{4 \cdot \frac{1}{5}} = 10^{\frac{4}{5}} \][/tex]
Now, we multiply these two expressions together:
[tex]\[ 10^{\frac{3}{5}} \cdot 10^{\frac{4}{5}} \][/tex]
Using the property of exponents that states [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex]:
[tex]\[ 10^{\frac{3}{5}} \cdot 10^{\frac{4}{5}} = 10^{\frac{3}{5} + \frac{4}{5}} = 10^{\frac{3+4}{5}} = 10^{\frac{7}{5}} \][/tex]
Thus, the expression [tex]\( \sqrt[5]{10^3} \cdot \sqrt[5]{10^4} \)[/tex] can be simplified to:
[tex]\[ 10^{\frac{7}{5}} \][/tex]
Now, we can express [tex]\( 10^{\frac{7}{5}} \)[/tex] in a more straightforward decimal form if needed. The exponent [tex]\( \frac{7}{5} \)[/tex] is equivalent to 1.4. So, we have:
[tex]\[ 10^{1.4} \approx 25.118864315095795 \][/tex]
So, the given expression [tex]\( \sqrt[5]{10^3} \cdot \sqrt[5]{10^4} \)[/tex] is equivalent to:
[tex]\[ 10^{\frac{7}{5}} \text{ or approximately } 25.118864315095795 \][/tex]
