To evaluate or simplify the expression [tex]\(5^{-7}\)[/tex] without negative exponents, follow these steps:
1. Understand the rule of negative exponents:
According to the rules of exponents, [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex]. This means that any number raised to a negative exponent is equal to the reciprocal of that number raised to the corresponding positive exponent.
2. Apply the rule to the given expression:
Given [tex]\(5^{-7}\)[/tex], we can rewrite this using the rule for negative exponents.
[tex]\[
5^{-7} = \frac{1}{5^7}
\][/tex]
3. Calculate the value of [tex]\(5^7\)[/tex]:
To express the original expression without negative exponents, we calculate [tex]\(5^7\)[/tex]:
[tex]\[
5^7 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5
\][/tex]
4. Evaluate the fraction:
The expression can now be evaluated as follows:
[tex]\[
5^{-7} = \frac{1}{5^7}
\][/tex]
Given that [tex]\(5^7 \approx 78125\)[/tex], the expression becomes:
[tex]\[
\frac{1}{78125}
\][/tex]
This gives us the simplified form of the expression without negative exponents.
5. Express in scientific notation for simplicity:
The simplified value [tex]\(\frac{1}{78125}\)[/tex] can also be expressed in scientific notation:
[tex]\[
\frac{1}{78125} \approx 1.28 \times 10^{-5}
\][/tex]
Therefore, the simplified form of [tex]\(5^{-7}\)[/tex] without negative exponents is [tex]\(\frac{1}{78125}\)[/tex], and its approximate decimal value is [tex]\(1.28 \times 10^{-5}\)[/tex].
