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# Find the value: a) ${2^{10}}$ b) ${5^3}$c) ${\left( { - 7} \right)^3}$d) ${8^1}$

Last updated date: 13th Jun 2024
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Hint: When a whole number ‘n’ and a real number ‘x’ represented as ${x^n}$ is the repetitive multiplication of real number ‘x’, n times ${x^n} = x \times x \times x \times x \times x..........n - times$also known as “ x raised to the power n” where x is the base and n is the exponents or the power.
If the value of the exponent is, $n = 2$ it is known as the square of a number, or when a number is multiplied with itself, then the resultant is known as the squared number (${x^2} = x \times x$).
If the value of the exponent is, $n = 3$ it is known as the cube of a number, or when a number is multiplied by itself two times, then the result is a cubic number (${x^3} = x \times x \times x$).
In the question, we need to determine the numeric value of different exponential terms (i.e., raised to the power terms) for which we have used the algebraic calculations only.

Complete step by step answer:
(i)For the given number, ${2^{10}}$ we can see the number has the exponents power is 10 whereas 2 is its base hence we can find the value as:
${2^{10}} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 1024$
Hence, ${2^{10}} = 1024$.
(ii)
Now for ${5^3}$we have to find the cube of the number as $n = 3$here, so
${5^3} = 5 \times 5 \times 5 = 125$
Hence, the value of ${5^3} = 125$.
(iii)
Now, for ${\left( { - 7} \right)^3}$we have to find the cube of the number, we can write
${\left( { - 7} \right)^3} = {\left( { - 1} \right)^3}{\left( 7 \right)^3} = {\left( { - 1} \right)^3}\left\{ {7 \times 7 \times 7} \right\} = - 343$
Hence, ${\left( { - 7} \right)^3} = - 343$.
(iv)
Now for the number ${8^1}$we can see, there is the exponential power of the number is 1, i.e. n=1 which means there is no power to the number hence we can write ${8^1} = 8$.
Hence, ${8^1} = 8$

Note: When an odd number of the negative integer is multiplied, then the result is also a negative integer ${\left( { - 1} \right)^3} = \left( { - 1} \right) \times \left( { - 1} \right) \times \left( { - 1} \right) = - 1$.