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1. Use the Power Rule of Exponents to simplify the expression (6^2)^4

2. Use the Power Rule of Exponents to simplify the expression (73^4)^5

3. Use the Power Rule of Exponents to simplify the expression (78^8)^6

4. Use the Power Rule of Exponents to create a simplified equivalent expression of (12 x 12 x 12) x (12x12x12)

5. What exponent would make the two expressions equivalent: (16^x)^6 = (16^18)

5 answers

Let's simplify each expression using the Power Rule of Exponents, which states that \((a^m)^n = a^{m \cdot n}\).

  1. For the expression \((6^2)^4\): \[ (6^2)^4 = 6^{2 \cdot 4} = 6^8 \]

  2. For the expression \((73^4)^5\): \[ (73^4)^5 = 73^{4 \cdot 5} = 73^{20} \]

  3. For the expression \((78^8)^6\): \[ (78^8)^6 = 78^{8 \cdot 6} = 78^{48} \]

  4. To simplify the expression \((12 \times 12 \times 12) \times (12 \times 12 \times 12)\), we can count the total number of 12's: \[ (12^3) \times (12^3) = 12^{3 + 3} = 12^6 \]

  5. For the equation \((16^x)^6 = (16^{18})\), we use the Power Rule: \[ 16^{x \cdot 6} = 16^{18} \] This implies: \[ x \cdot 6 = 18 \] Now, solve for \(x\): \[ x = \frac{18}{6} = 3 \]

In summary:

  1. \(6^8\)
  2. \(73^{20}\)
  3. \(78^{48}\)
  4. \(12^6\)
  5. \(x = 3\)
Which expression is correctly developed to use the Power Rule of Exponents?
(7^4)^8
12^9
(7^8) x (7^10)
(-5)^0

What exponent would make the two expressions equivalent: (3^3)^x = 3^24

Let's analyze each expression to determine if they are correctly developed to use the Power Rule of Exponents.

  1. Expression: \((7^4)^8\)

    • This expression can be simplified using the Power Rule, as follows:
      \((7^4)^8 = 7^{4 \cdot 8} = 7^{32}\)
    • Correctly developed.

  2. Expression: \(12^9\)

    • This expression does not apply the Power Rule, as it is already expressed in simplified form.
    • Not applicable for the Power Rule.

  3. Expression: \((7^8) \times (7^{10})\)

    • This expression combines two bases with the same base, which can be simplified as follows:
      \(7^8 \times 7^{10} = 7^{8 + 10} = 7^{18}\)
    • Correctly developed.

  4. Expression: \((-5)^0\)

    • By the Zero Exponent Rule, any non-zero number raised to the power of zero equals 1. This does not use the Power Rule but rather applies the zero exponent property.
    • Not applicable for the Power Rule.

Summary of Correctly Developed Expressions:

  • \((7^4)^8\) is correctly developed.
  • \((7^8) \times (7^{10})\) is correctly developed.
  • \(12^9\) and \((-5)^0\) do not apply the Power Rule in any transformation.

Now, for the given exponential equation:

To solve: \((3^3)^x = 3^{24}\)

Using the Power Rule: \[ (3^3)^x = 3^{3x} \] Set the exponents equal to each other: \[ 3x = 24 \] Now, divide both sides by 3: \[ x = \frac{24}{3} = 8 \]

The exponent that makes the two expressions equivalent is:

\(x = 8\)

1. (7^4)^8

2. 10^63

3. (4^5)^2

4. 15^12

5. 8

Let's analyze each expression in relation to the Power Rule of Exponents, which states that \((a^m)^n = a^{m \cdot n}\).

1. \((7^4)^8\)

Using the Power Rule: \[ (7^4)^8 = 7^{4 \cdot 8} = 7^{32} \]

  • This is correctly developed.

2. \(10^{63}\)

This expression does not apply the Power Rule directly, as it is already in its simplest form.

  • Not applicable for the Power Rule but is correct in its current form.

3. \((4^5)^2\)

Using the Power Rule: \[ (4^5)^2 = 4^{5 \cdot 2} = 4^{10} \]

  • This is correctly developed.

4. \(15^{12}\)

Similar to \(10^{63}\), this expression is in its simplest form and does not apply the Power Rule.

  • Not applicable for the Power Rule but is correct as it stands.

5. \(8\)

This expression does not make use of exponents in any form. However, if intended, it can be expressed as \(8^1\) (which does not apply the Power Rule).

  • Not applicable for the Power Rule.

Summary:

The expressions that correctly use the Power Rule of Exponents are:

  • \((7^4)^8\) simplifies to \(7^{32}\).
  • \((4^5)^2\) simplifies to \(4^{10}\).
  • Both \(10^{63}\) and \(15^{12}\) are already defined correctly but do not illustrate the Power Rule since they are not products of exponents that need simplification.

Overall, the final results from the original equations are:

  1. \(7^{32}\)
  2. \(10^{63}\) (no simplification needed)
  3. \(4^{10}\)
  4. \(15^{12}\) (no simplification needed)
  5. \(8\) (but can be expressed as \(8^1\) if necessary).