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Play the property of negative exponents to generate an expression equivalent to 3^-17/8^-14 with positive exponents only

3^-17/8^-14 = blank

11 answers

To simplify the expression \( \frac{3^{-17}}{8^{-14}} \) using the property of negative exponents, we can rewrite the negative exponents as follows:

\[ \frac{3^{-17}}{8^{-14}} = 3^{-17} \cdot 8^{14} \]

Next, we need to express \( 8^{14} \) using base 2, since \( 8 = 2^3 \):

\[ 8^{14} = (2^3)^{14} = 2^{3 \cdot 14} = 2^{42} \]

Now the original expression becomes:

\[ 3^{-17} \cdot 8^{14} = 3^{-17} \cdot 2^{42} \]

Next, we can express \( 3^{-17} \) as:

\[ 3^{-17} = \frac{1}{3^{17}} \]

So, the expression now looks like:

\[ \frac{2^{42}}{3^{17}} \]

Therefore, the final expression with positive exponents only is:

\[ \frac{2^{42}}{3^{17}} \]

Thus, your result is:

\[ \frac{2^{42}}{3^{17}} \]

R u sure

Let's carefully double-check the steps for converting \( \frac{3^{-17}}{8^{-14}} \) to an expression with only positive exponents.

  1. Start with the expression: \[ \frac{3^{-17}}{8^{-14}} \]

  2. Use the property of negative exponents: \[ \frac{3^{-17}}{8^{-14}} = 3^{-17} \cdot 8^{14} \]

  3. Rewrite \( 8 \) as \( 2^3 \): \[ 8^{14} = (2^3)^{14} = 2^{3 \cdot 14} = 2^{42} \]

  4. Now substitute this back into the expression: \[ 3^{-17} \cdot 8^{14} = 3^{-17} \cdot 2^{42} \]

  5. Next, rewrite \( 3^{-17} \): \[ 3^{-17} = \frac{1}{3^{17}} \]

  6. Therefore, combining everything gives: \[ \frac{2^{42}}{3^{17}} \]

So, the correct final expression with only positive exponents is indeed: \[ \frac{2^{42}}{3^{17}} \]

Yes, I am confident in this result. If you have any further questions or if there's anything specific you would like to clarify, please let me know!

Apply the properties of integer exponents to generate equivalent expressions to (7^-3) ^7 * 7^20/ 6^-2 with only positive exponents, then solve the expression. Provide the answer as an improper fraction.

(7^-3) ^7 * 7^20/ 6^-2 = blank

To simplify the expression \( \frac{(7^{-3})^7 \cdot 7^{20}}{6^{-2}} \) and convert it into an equivalent expression with only positive exponents, let's follow these steps:

  1. First, simplify \( (7^{-3})^7 \):
    Using the property of exponents \( (a^m)^n = a^{m \cdot n} \): \[ (7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21} \]

  2. Now multiply by \( 7^{20} \): \[ 7^{-21} \cdot 7^{20} = 7^{-21 + 20} = 7^{-1} \]

  3. Now we have the current expression: \[ \frac{7^{-1}}{6^{-2}} \]

  4. Convert the negative exponent in the denominator: \[ \frac{7^{-1}}{6^{-2}} = 7^{-1} \cdot 6^2 \]

  5. Now apply the property of negative exponents again: \[ 7^{-1} = \frac{1}{7} \] Therefore: \[ 7^{-1} \cdot 6^2 = \frac{6^2}{7} = \frac{36}{7} \]

So, the simplified expression with only positive exponents is: \[ \frac{36}{7} \]

Final Answer

The equivalent expression is: \[ \frac{36}{7} \]

This is in the form of an improper fraction as requested. If you have any further questions or need any additional assistance, feel free to ask!

So its 36/7?

Yes, that's correct! The expression

\[ \frac{(7^{-3})^7 \cdot 7^{20}}{6^{-2}} \]

simplifies to

\[ \frac{36}{7} \]

This is an improper fraction, and it is indeed the final answer. If you have any more questions or need further clarification, feel free to ask!

Simplify the expression(4x^3)^2 using the properties of integer exponents.

4x^6

16x^6

8x^6

16x^5

To simplify the expression \((4x^3)^2\) using the properties of integer exponents, we can apply the following rules:

  1. Power of a product: \((ab)^n = a^n b^n\)
  2. Power of a power: \((a^m)^n = a^{m \cdot n}\)

Now, applying these rules to the expression:

\[ (4x^3)^2 = 4^2 \cdot (x^3)^2 \]

Calculating each part:

  • \(4^2 = 16\)
  • \((x^3)^2 = x^{3 \cdot 2} = x^6\)

Now we combine these results:

\[ (4x^3)^2 = 16x^6 \]

So, the simplified expression is:

\[ \boxed{16x^6} \]

Thus, the correct answer is 16x^6.

Play the properties of integer exponents to generate an equivalent, algebraic expression to (4x^4 y ^-4)^3/ (x^7y3)^2

To simplify the expression

\[ \frac{(4x^4 y^{-4})^3}{(x^7 y^3)^2}, \]

we will apply the properties of integer exponents step-by-step.

Step 1: Simplify the numerator

Start with the numerator:

\[ (4x^4 y^{-4})^3. \]

Using the property of exponents \((a^m)^n = a^{mn}\) and the power of a product property, we get:

\[ = 4^3 (x^4)^3 (y^{-4})^3 = 64 x^{12} y^{-12}. \]

Step 2: Simplify the denominator

Now, simplify the denominator:

\[ (x^7 y^3)^2. \]

Using the same exponent properties, we have:

\[ = (x^7)^2 (y^3)^2 = x^{14} y^6. \]

Step 3: Combine the fractions

Now we can substitute these results back into the expression:

\[ \frac{64 x^{12} y^{-12}}{x^{14} y^6}. \]

Step 4: Simplify the fraction

Now we will simplify this fraction using the properties of exponents:

\[ = 64 \cdot \frac{x^{12}}{x^{14}} \cdot \frac{y^{-12}}{y^6}. \]

For the \(x\) terms, use \( \frac{a^m}{a^n} = a^{m-n} \):

\[ = 64 \cdot x^{12-14} \cdot y^{-12-6} = 64 \cdot x^{-2} \cdot y^{-18}. \]

Step 5: Rewrite with positive exponents

To express this with only positive exponents, recall that \(a^{-n} = \frac{1}{a^n}\):

\[ = \frac{64}{x^2 y^{18}}. \]

Thus, the final simplified equivalent algebraic expression is:

\[ \boxed{\frac{64}{x^2 y^{18}}}. \]