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Apply the zero exponent rule.

Mathematics

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3 parts
d
1
e

25633\frac{256\sqrt{3}}{3}

g
125

Step-by-step solution

10 steps

Here are the solutions to the simplification problems.

Part d· 2 steps

Simplify M0M^0

  1. Apply the zero exponent rule.

    Any non-zero number raised to the power of zero is equal to 1.

  2. Apply the rule to the expression.

    M0=1(assuming M≠0)M^0 = 1 \quad (\text{assuming } M \neq 0)

    The simplified expression is 1\boxed{1}.
    The statement "M0>1M^0 > 1" is incorrect, as M0M^0 is equal to 1.

Part e

25633\frac{256\sqrt{3}}{3}

Simplify 25×23(1+2)\frac{2^5 \times 2^3}{\sqrt{(1+2)}}

  1. Simplify the numerator using the product rule for exponents (am×an=am+na^m \times a^n = a^{m+n}).

    25×23=25+3=282^5 \times 2^3 = 2^{5+3} = 2^8

  2. Calculate the value of 282^8.

    28=2562^8 = 256

  3. Simplify the denominator.

    (1+2)=3\sqrt{(1+2)} = \sqrt{3}

  4. Combine the simplified numerator and denominator.

    2563\frac{256}{\sqrt{3}}

  5. Rationalize the denominator by multiplying the numerator and denominator by 3\sqrt{3}.

    2563×33=25633\frac{256}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{256\sqrt{3}}{3}

    The simplified expression is 25633\boxed{\frac{256\sqrt{3}}{3}}.

Part g125

Simplify (8−3)3(8-3)^3

  1. Perform the subtraction inside the parentheses.

    8−3=58-3 = 5

  2. Raise the result to the power of 3.

    53=5×5×55^3 = 5 \times 5 \times 5

  3. Calculate the product.

    5×5×5=25×5=1255 \times 5 \times 5 = 25 \times 5 = 125

    The simplified expression is 125\boxed{125}.
    The statement "> 125" is incorrect, as the result is exactly 125.

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