Question

Find the value of $4 \times {64^{ - \dfrac{1}{2}}}\left( {{{64}^{\dfrac{1}{2}}} + {{64}^{\dfrac{3}{2}}}} \right)$.

Answer 1

Hint: First move the negative power to the denominator to make the power positive. Then find the square root of the values. After that take-out common from the values in the bracket and cancel out the common factors from the numerator and denominator. Now, simplify the values in the bracket and then multiply the terms. The value that comes out will be the desired result.

Complete step-by-step answer:
As we know that, ${x^{ - a}} = \dfrac{1}{{{x^a}}}$,
$4 \times {64^{ - \dfrac{1}{2}}}\left( {{{64}^{\dfrac{1}{2}}} + {{64}^{\dfrac{3}{2}}}} \right) = \dfrac{{4\left( {{{64}^{\dfrac{1}{2}}} + {{64}^{\dfrac{3}{2}}}} \right)}}{{{{64}^{ - \dfrac{1}{2}}}}}$
Also, ${x^{\dfrac{1}{n}}} = \sqrt[n]{x}$, then,
$4 \times {64^{ - \dfrac{1}{2}}}\left( {{{64}^{\dfrac{1}{2}}} + {{64}^{\dfrac{3}{2}}}} \right) = \dfrac{{4\left[ {\sqrt {64} + {{\left( {\sqrt {64} } \right)}^3}} \right]}}{{\sqrt {64} }}$
Find the square root of the values,
$4 \times {64^{ - \dfrac{1}{2}}}\left( {{{64}^{\dfrac{1}{2}}} + {{64}^{\dfrac{3}{2}}}} \right) = \dfrac{{4\left[ {8 + {{\left( 8 \right)}^3}} \right]}}{8}$
Take out common from the bracket in the numerator,
$4 \times {64^{ - \dfrac{1}{2}}}\left( {{{64}^{\dfrac{1}{2}}} + {{64}^{\dfrac{3}{2}}}} \right) = \dfrac{{4 \times 8\left[ {1 + {{\left( 8 \right)}^2}} \right]}}{8}$
Cancel out the common factors from the numerator and denominator,
$4 \times {64^{ - \dfrac{1}{2}}}\left( {{{64}^{\dfrac{1}{2}}} + {{64}^{\dfrac{3}{2}}}} \right) = 4\left( {1 + {8^2}} \right)$
Square the term and add 1 in it,
$4 \times {64^{ - \dfrac{1}{2}}}\left( {{{64}^{\dfrac{1}{2}}} + {{64}^{\dfrac{3}{2}}}} \right) = 4 \times 65$
Multiply the terms on the right to get the desired result,
$4 \times {64^{ - \dfrac{1}{2}}}\left( {{{64}^{\dfrac{1}{2}}} + {{64}^{\dfrac{3}{2}}}} \right) = 260$

Hence, the value of $4 \times {64^{ - \dfrac{1}{2}}}\left( {{{64}^{\dfrac{1}{2}}} + {{64}^{\dfrac{3}{2}}}} \right)$ is 260.

Note: An expression that represents repeated multiplication of the same factor is called a power.
Multiplication Rule: If two powers have the same base then we can multiply the powers. When we multiply two powers we add their exponents.
${x^a} \times {x^b} = {x^{a + b}}$
Division Rule: If two powers have the same base then we can divide the powers. When we divide powers we subtract their exponents.
$\dfrac{{{x^a}}}{{{x^b}}} = {x^{a - b}}$
Negative Exponent: A negative exponent is the same as the reciprocal of the positive exponent.
${x^{ - a}} = \dfrac{1}{{{x^a}}}$
Power of Power Rule: The rule for the power of a power and the power of a product can be combined into the following rule:
${\left( {{x^a} \times {y^b}} \right)^z} = {x^{az}} \times {y^{bz}}$
Zero raised to any power is zero (e.g. ${0^5} = 0$).
One raised to any power is one (e.g. ${1^5} = 1$).
Any number raised to the zero power is one (e.g. ${7^0} = 1$).
Any number raised to the first power is that number (e.g. ${7^1} = 7$).