Answer :
To find the expanded form of the exponential expression [tex]\((10^{-4})^3\)[/tex], let's break it down step-by-step:
1. Understand the Exponential Expression:
- The expression [tex]\((10^{-4})^3\)[/tex] involves raising the power of [tex]\(-4\)[/tex] to another power, [tex]\(3\)[/tex].
2. Apply the Power Rule:
- When you have an expression [tex]\((a^m)^n\)[/tex], you can simplify it using the power rule: [tex]\((a^m)^n = a^{m \cdot n}\)[/tex].
- In this case, [tex]\(a = 10\)[/tex], [tex]\(m = -4\)[/tex], and [tex]\(n = 3\)[/tex].
3. Multiply the Exponents:
- Calculate the new exponent by multiplying [tex]\(-4\)[/tex] and [tex]\(3\)[/tex]:
[tex]\[
-4 \times 3 = -12
\][/tex]
4. Rewrite the Expression with the New Exponent:
- The expression [tex]\((10^{-4})^3\)[/tex] becomes [tex]\(10^{-12}\)[/tex].
5. Convert to Expanded Form:
- [tex]\(10^{-12}\)[/tex] means the reciprocal of [tex]\(10^{12}\)[/tex], which is:
[tex]\[
\frac{1}{10^{12}}
\][/tex]
- This can be written as [tex]\(1 \times 10^{-12}\)[/tex] or in decimal form as [tex]\(1 \times 10^{-12} = 0.000000000001\)[/tex].
So, the expanded form of [tex]\((10^{-4})^3\)[/tex] is [tex]\(\frac{1}{10^{12}}\)[/tex] or [tex]\(1 \times 10^{-12}\)[/tex].
1. Understand the Exponential Expression:
- The expression [tex]\((10^{-4})^3\)[/tex] involves raising the power of [tex]\(-4\)[/tex] to another power, [tex]\(3\)[/tex].
2. Apply the Power Rule:
- When you have an expression [tex]\((a^m)^n\)[/tex], you can simplify it using the power rule: [tex]\((a^m)^n = a^{m \cdot n}\)[/tex].
- In this case, [tex]\(a = 10\)[/tex], [tex]\(m = -4\)[/tex], and [tex]\(n = 3\)[/tex].
3. Multiply the Exponents:
- Calculate the new exponent by multiplying [tex]\(-4\)[/tex] and [tex]\(3\)[/tex]:
[tex]\[
-4 \times 3 = -12
\][/tex]
4. Rewrite the Expression with the New Exponent:
- The expression [tex]\((10^{-4})^3\)[/tex] becomes [tex]\(10^{-12}\)[/tex].
5. Convert to Expanded Form:
- [tex]\(10^{-12}\)[/tex] means the reciprocal of [tex]\(10^{12}\)[/tex], which is:
[tex]\[
\frac{1}{10^{12}}
\][/tex]
- This can be written as [tex]\(1 \times 10^{-12}\)[/tex] or in decimal form as [tex]\(1 \times 10^{-12} = 0.000000000001\)[/tex].
So, the expanded form of [tex]\((10^{-4})^3\)[/tex] is [tex]\(\frac{1}{10^{12}}\)[/tex] or [tex]\(1 \times 10^{-12}\)[/tex].
