Questions: y^8 * y^-2 = y^8 / 1 * 1 / y^2 = 1 * y^4 / y^5 = y^-1 = 1 / y (x^4)^-3 = x^-12 = 1 / x^12 (65^4 t^-7)^2 = 36 s^8 / 1 (4 / x^3)^-3 = 6

y^8 * y^-2 = y^8 / 1 * 1 / y^2 = 1 * y^4 / y^5 = y^-1 = 1 / y
(x^4)^-3 = x^-12 = 1 / x^12
(65^4 t^-7)^2 = 36 s^8 / 1
(4 / x^3)^-3 = 6
Transcript text: $y^{8} \cdot y^{-2}=\frac{y^{8}}{1} \cdot \frac{1}{y^{2}}=$ $1 . \frac{y^{4}}{y^{5}}=y^{-1}=\frac{1}{y}$ $\left(x^{4}\right)^{-3}=x^{-12}=\frac{1}{x^{12}}$ $\left(65^{4} t^{-7}\right)^{2}=\frac{36 s^{8}}{1}$ $\left(\frac{4}{x^{3}}\right)^{-3}=6$
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Solution

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Solution Steps

To solve the first three questions, we will use the properties of exponents.

  1. For the expression \( y^8 \cdot y^{-2} \), we will apply the rule that states when multiplying like bases, you add the exponents.
  2. For the expression \( \left(x^4\right)^{-3} \), we will use the power of a power rule, which involves multiplying the exponents.
  3. For the expression \( \left(65^4 t^{-7}\right)^2 \), we will apply the power of a product rule, which involves distributing the exponent to each factor inside the parentheses.
Step 1: Simplifying \( y^8 \cdot y^{-2} \)

Using the property of exponents that states \( a^m \cdot a^n = a^{m+n} \), we can simplify: \[ y^8 \cdot y^{-2} = y^{8 + (-2)} = y^6 \]

Step 2: Simplifying \( \left(x^4\right)^{-3} \)

Applying the power of a power rule, which states \( (a^m)^n = a^{m \cdot n} \), we have: \[ \left(x^4\right)^{-3} = x^{4 \cdot (-3)} = x^{-12} \]

Step 3: Simplifying \( \left(65^4 t^{-7}\right)^2 \)

Using the power of a product rule, which states \( (ab)^n = a^n b^n \), we can distribute the exponent: \[ \left(65^4 t^{-7}\right)^2 = 65^{4 \cdot 2} \cdot t^{-7 \cdot 2} = 65^8 \cdot t^{-14} \]

Final Answer

The results for each expression are:

  1. \( y^6 \)
  2. \( x^{-12} \)
  3. \( 65^8 \cdot t^{-14} \)

Thus, the final answers are: \[ \boxed{y^6}, \quad \boxed{x^{-12}}, \quad \boxed{65^8 \cdot t^{-14}} \]

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