Question

Use properties of rational exponents to simplify the expression. Assume that all variables represent positive numbers.
x Superscript 1 divided by 9 Baseline times x Superscript 1 divided by 7
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Part 1
x Superscript 1 divided by 9 Baseline times x Superscript 1 divided by 7equals
  
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Answer 1

To simplify the expression \( x^{\frac{1}{9}} \times x^{\frac{1}{7}} \), we can use the property of exponents that states:

\[
x^a \times x^b = x^{a+b}
\]

In this case, we have:

\[
a = \frac{1}{9} \quad \text{and} \quad b = \frac{1}{7}
\]

So we need to add these two fractions:

\[
\frac{1}{9} + \frac{1}{7}
\]

To add these fractions, we need a common denominator. The least common multiple of 9 and 7 is 63. We convert both fractions:

\[
\frac{1}{9} = \frac{7}{63}
\]
\[
\frac{1}{7} = \frac{9}{63}
\]

Now, we can add them:

\[
\frac{7}{63} + \frac{9}{63} = \frac{16}{63}
\]

Now we can write the expression:

\[
x^{\frac{1}{9}} \times x^{\frac{1}{7}} = x^{\frac{16}{63}}
\]

Thus, the simplified expression is:

\[
\boxed{x^{\frac{16}{63}}}
\]