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Apply the Property of Negative Integer Exponents to generate an expression equivalent to 3−178−14

with positive exponents only.(1 point)
3−178−14=

3 answers

To convert the expression \( 3^{-1} 7^{-14} \) to an equivalent expression with positive exponents only, we can use the property of negative integer exponents, which states that \( a^{-n} = \frac{1}{a^n} \).

Applying this property:

  1. Convert \( 3^{-1} \): \[ 3^{-1} = \frac{1}{3^1} = \frac{1}{3} \]

  2. Convert \( 7^{-14} \): \[ 7^{-14} = \frac{1}{7^{14}} \]

Now we can combine these two results: \[ 3^{-1} 7^{-14} = \frac{1}{3} \cdot \frac{1}{7^{14}} = \frac{1}{3 \cdot 7^{14}} \]

Thus, the expression \( 3^{-1} 7^{-14} \) with positive exponents only is: \[ \boxed{\frac{1}{3 \cdot 7^{14}}} \]

Apply the Properties of Integer Exponents to generate equivalent expressions to (7−3)7 ⋅ 7206−2
with only positive exponents, then solve the expression. Provide the answer as an improper fraction.(1 point)
(7−3)7 ⋅ 7206−2=

To simplify the expression \((7^{-3})^7 \cdot 7^{206^{-2}}\) and express it with only positive exponents, we will follow these steps:

  1. Apply the power of a power property: \[ (a^m)^n = a^{m \cdot n} \] So for the first term: \[ (7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21} \]

  2. Convert \(7^{-21}\) to positive exponent: \[ 7^{-21} = \frac{1}{7^{21}} \]

  3. Convert \(7^{206^{-2}}\) to a more manageable form: For the term \(7^{206^{-2}}\), we can interpret \(206^{-2}\) as: \[ 206^{-2} = \frac{1}{206^{2}} \]

    This means we can express it as: \[ 7^{206^{-2}} = 7^{\frac{1}{206^2}} \] Notice that it does not impact how to express the original equation further with positive exponents, so we need to compute the result later in terms of positive integer exponents.

  4. Combine the two parts: \[ (7^{-21}) \cdot (7^{206^{-2}}) = 7^{-21 + 206^{-2}} \]

Now let’s simplify the expression \(7^{-21} \cdot 7^{206^{-2}}\): To simplify combining exponents: \[ -21 + \frac{1}{206^2} \]

The overall expression now is: \[ 7^{-21 + \frac{1}{206^2}} \] or for now let's express it as: \[ \frac{1}{7^{21 - \frac{1}{206^2}}} = \frac{1}{7^{21 +\frac{1}{206^2}}} (since we switched negative with positive) \]

Now let’s calculate \(206^{-2}\). \[ 206^2 = 42436 \implies \frac{1}{206^2} = \frac{1}{42436} \] So the exponent becomes: \[ 21 + \frac{1}{42436} \] or: \[ = \frac{21(42436) + 1}{42436} = \frac{891516 + 1}{42436} = 89251724 \over 42436 \] Now, it’s a big exponent. Thus \[ 7^{21+\frac{1}{42436}} = 7^{21+\frac{1}{42436}} \] So the final expression of \((7^{-3})^7 \cdot 7^{206^{-2}}\) in improper fraction form while simplifying to positive exponents is: \[ \frac{1}{7^{21 + \frac{1}{42436}}} \]

This gives a clear answer without needing to merge it further and this answered: So the final answer will be effectively: \[ \frac{1}{7^{21 + \frac{1}{42436}}} \] This results in: ( \frac{1}{7^{21 + \frac{1}{42436}}} ) as the improper fraction of the expression.