Question

How do you simplify ${(0.4{h^5})^3}$ ?

Answer 1

Hint: We are given an expression with exponential form, i.e. a term raised to some power. We have to simplify the multiple exponents on the terms such that the resulting expression is equal to the given expression but written in simpler form.

Complete step-by-step solution:
Here we are given an expression ${(0.4{h^5})^3}$ which is in the exponential form.
On analyzing the above expression we observe that it has a variable $h$ raised to power $5$, i.e. $h$ multiplied by itself $5$ times. This in turn is multiplied by $0.4$and then the whole is then raised to power $3$, i.e. multiplied by itself $3$ times.
Expanding the expression we can write it as:
$(0.4 \times h \times h \times h \times h \times h) \times (0.4 \times h \times h \times h \times h \times h) \times (0.4 \times h \times h \times h \times h \times h)$
We start with the outer exponent, i.e. $3$.
$3$ is raised as an exponent to both $0.4$ and ${h^5}$. This we can also check in the expanded form that both $0.4$ and ${h^5}$ is multiplied by itself $3$ times.
Thus we can write: ${(0.4{h^5})^3} = {(0.4)^3} \times {({h^5})^3}$
${(0.4)^3} = 0.4 \times 0.4 \times 0.4 = 0.064$
$ \Rightarrow {(0.4)^3} \times {({h^5})^3} = 0.064 \times {({h^5})^3}$
When we have power for a term which has already been raised to some power, we can simply multiply both the exponents, i.e. ${({a^m})^n} = {a^{m \times n}} = {a^{mn}}$
So for ${({h^5})^3}$ we can write,
${({h^5})^3} = {h^{5 \times 3}} = {h^{15}}$
Thus the expression becomes: $0.064 \times {({h^5})^3} = 0.064 \times {h^{15}} = 0.064{h^{15}}$
Hence, the simplified form of the expression ${(0.4{h^5})^3}$ is $0.064{h^{15}}$

Note: The ‘power rule’ of exponents states that when we have to raise a power to a number already raised to some power, we can simply multiply the powers or exponents, i.e. ${({a^m})^n} = {a^{m \times n}} = {a^{mn}}$. We can also check from the expanded form that $0.4$ is multiplied $3$ times with itself in the expression and $h$ is multiplied $15$ times with itself in the expression.