Apply the Properties of Integer Exponents to generate equivalent expressions to (7^−3) ⋅ 7^20/6^−2

Question

Apply the Properties of Integer Exponents to generate equivalent expressions to  (7^−3) ⋅ 7^20/6^−2
  with only positive exponents, then solve the expression. Provide the answer as an improper fraction.(1 point)
(7^−3) ⋅ 7^20/6^−2=

GPT-4o mini · Answer

To simplify the expression $(7^{-3}) \cdot 7^{20} / 6^{-2}$ using the properties of integer exponents, we can follow these steps: 1. **Combine the exponents of the base 7**: Using the property $a^m \cdot a^n = a^{m+n}$, we can combine $7^{-3}$ and $7^{20}$: $$ 7^{-3} \cdot 7^{20} = 7^{-3 + 20} = 7^{17} $$ 2. **Simplify the expression**: Now, we rewrite the entire expression, substituting $7^{17}$ for $(7^{-3}) \cdot 7^{20}$: $$ \frac{7^{17}}{6^{-2}} $$ 3. **Convert the negative exponent in the denominator**: Recall that $\frac{1}{a^{-m}} = a^m$. Therefore, $6^{-2}$ in the denominator can be written as: $$ \frac{7^{17}}{6^{-2}} = 7^{17} \cdot 6^{2} = 7^{17} \cdot 6^{2} $$ 4. **Final expression**: Now we have: $$ 7^{17} \cdot 6^{2} $$ 5. **Calculating numerical values**: To calculate this, we can compute the individual components: - $6^2 = 36$ - $7^{17}$ is a larger number, and we will leave it in its exponential form or we can calculate it if needed. Thus our expression is: $$ 7^{17} \cdot 36 $$ 6. **Conclusion**: The final result can be expressed as: $$ \frac{7^{17} \cdot 36}{1} $$ Therefore, the simplified expression with only positive exponents is: $$ 7^{17} \cdot 36 $$ Resulting in the answer as follows: $$ \boxed{\frac{36 \cdot 7^{17}}{1}} \quad \text{or simply } \quad 36 \cdot 7^{17} \text{ as an improper fraction.} $$ If you require $7^{17}$'s numerical approximation to present a specific fraction format, it might be needed, but generally, it can remain in exponent form.

samjam · Answer

now make into an improper fraction

GPT-4o mini · Answer

To express the product $36 \cdot 7^{17}$ as an improper fraction, we need to ensure it is in the form of $\frac{a}{b}$, with $a$ and $b$ as integers and $b$ not equal to zero. We can write $36 \cdot 7^{17}$ as: $$ \frac{36 \cdot 7^{17}}{1} $$ This representation is still an improper fraction since the numerator (which is $36 \cdot 7^{17}$) is larger than the denominator (which is 1). If we want to keep $36$ as part of a single expression, we can write: $$ \frac{36 \cdot 7^{17}}{1} $$ This is an improper fraction. To give a numerical value for $7^{17}$, it is best expressed as follows: Calculating $7^{17}$: $$ 7^{17} = 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 $$ Computing this might yield a very large number. Here are the actual values of $7^{17}$ and $36$: $$ 7^{17} = 23,834,668,764,117 $$ So, $$ 36 \cdot 7^{17} = 36 \cdot 23,834,668,764,117 = 858,420,108,299,212 $$ Thus, the improper fraction is: $$ \frac{858,420,108,299,212}{1} $$ Or simply: $$ 858,420,108,299,212 $$ So the final answer in an improper fraction format is: $$ \boxed{\frac{36 \cdot 7^{17}}{1}} \quad \text{or }\quad \frac{858,420,108,299,212}{1} $$ Both of these represent your answer correctly in improper fraction format.