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Is the equation (8–√4)−24⋅(8–√)16=8−48

true or false? Select the response that correctly answers the question and provides the appropriate justification.(1 point)
Responses

False, because the exponents should be added rather than multiplied.
False, because the exponents should be added rather than multiplied.

True. When the bases are multiplied, the exponents are also multiplied.
True. When the bases are multiplied, the exponents are also multiplied.

False, because the Negative Exponent Rule should be applied.
False, because the Negative Exponent Rule should be applied.

True, but the equation can be simplified further.
True, but the equation can be simplified further.

1 answer

To determine whether the equation \((8−\sqrt{4})−24⋅(8−\sqrt{16})=8−48\) is true or false, let's simplify both sides of the equation step by step.

First, we evaluate the expressions on the left side:

  1. Calculate \(\sqrt{4}\) and \(\sqrt{16}\):

    • \(\sqrt{4} = 2\)
    • \(\sqrt{16} = 4\)

  2. Substituting the square roots into the expression:

    • \(8 - \sqrt{4} = 8 - 2 = 6\)
    • \(8 - \sqrt{16} = 8 - 4 = 4\)

  3. Now substitute values back into the left side:

    • The left side becomes: \[ (8 - \sqrt{4}) - 24 \cdot (8 - \sqrt{16}) = 6 - 24 \cdot 4 = 6 - 96 \]
    • Simplifying that gives: \[ 6 - 96 = -90 \]

Now, let's calculate the right side:

  1. Calculate \(8 - 48\): \[ 8 - 48 = -40 \]

Finally, we compare both sides:

  • Left side = -90
  • Right side = -40

Since -90 is not equal to -40, the equation is false.

Thus, the correct response is: False, because the exponents should be added rather than multiplied. (However, this is not the core reason for this particular equation being false; the equation is false due to the difference in results from both sides.)

To clarify, the response provided cannot be justified with "the exponents" context since it's not relevant to the equation as presented. The correct conclusion would indicate merely that the two sides yield different final results.

It might be better to select or provide an answer responsive to simply indicating the left-hand side evaluates to -90 and the right-hand side evaluates to -40.