To solve the equation [tex]\((-2)^{-3} \times (-2)^6 = (-2)^{2x-1}\)[/tex], let's follow these step-by-step instructions:
1. Simplify the left side of the equation using the properties of exponents:
We know that when you multiply two exponents with the same base, you can add the exponents together:
[tex]\[
(-2)^{-3} \times (-2)^6 = (-2)^{-3 + 6}
\][/tex]
Simplifying [tex]\(-3 + 6\)[/tex]:
[tex]\[
-3 + 6 = 3
\][/tex]
Therefore:
[tex]\[
(-2)^{-3} \times (-2)^6 = (-2)^3
\][/tex]
2. Evaluate the simplified left side:
Calculate [tex]\((-2)^3\)[/tex]:
[tex]\[
(-2)^3 = -2 \times -2 \times -2
\][/tex]
Performing the multiplications step-by-step:
[tex]\[
-2 \times -2 = 4 \quad \text{and then} \quad 4 \times -2 = -8
\][/tex]
Therefore:
[tex]\[
(-2)^{-3} \times (-2)^6 = -8
\][/tex]
3. Equate the simplified left side to the right side of the original equation:
Now, we have:
[tex]\[
(-2)^3 = (-2)^{2x-1}
\][/tex]
4. Since the bases are the same, set the exponents equal to each other:
[tex]\[
3 = 2x - 1
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Rearrange the equation to isolate [tex]\(x\)[/tex]:
[tex]\[
3 + 1 = 2x
\][/tex]
Simplify:
[tex]\[
4 = 2x
\][/tex]
Divide both sides by 2:
[tex]\[
x = 2
\][/tex]
6. Check the solution:
To confirm our solution, substitute [tex]\(x = 2\)[/tex] back into the right side of the equation:
[tex]\[
(-2)^{2x - 1} = (-2)^{2 \times 2 - 1} = (-2)^3
\][/tex]
Since [tex]\((-2)^3 = -8\)[/tex], we verify that the left side and right side of the equation are consistent.
Therefore, the solution to the equation [tex]\((-2)^{-3} \times (-2)^6 = (-2)^{2x-1}\)[/tex] is:
[tex]\[
x = 2
\][/tex]
