Simplifying Algebraic Expressions
The question asks for the value of the expression: $ \frac{(5.4)^3 - 0.064}{(5.4)^2 + 2.16 + 0.16} $
Recognizing Algebraic Identities
This expression resembles the algebraic identity for the difference of cubes:
$ a^3 - b^3 = (a - b)(a^2 + ab + b^2) $Let's identify the terms:
- Let $a = 5.4$. Then $a^3 = (5.4)^3$.
- We need to find $b$ such that $b^3 = 0.064$. Since $(0.4)^3 = 0.4 \times 0.4 \times 0.4 = 0.16 \times 0.4 = 0.064$, we have $b = 0.4$.
Verifying the Denominator
Now, let's check if the denominator matches the term $a^2 + ab + b^2$:
- $a^2 = (5.4)^2$
- $ab = (5.4)(0.4) = 2.16$
- $b^2 = (0.4)^2 = 0.16$
The denominator is $(5.4)^2 + 2.16 + 0.16$, which perfectly matches $a^2 + ab + b^2$.
Applying the Identity and Simplifying
Substitute $a = 5.4$ and $b = 0.4$ into the expression and apply the identity:
$ \frac{a^3 - b^3}{a^2 + ab + b^2} = \frac{(a - b)(a^2 + ab + b^2)}{a^2 + ab + b^2} $The term $(a^2 + ab + b^2)$ cancels out, leaving:
$ a - b $Calculating the Final Value
Substitute the values of $a$ and $b$ back:
$ 5.4 - 0.4 = 5 $Therefore, the value of the expression is 5.
Similar Questions
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Important Questions from Simplification
What will come in the place of the question mark ‘?’ in the following question?
(15 + 24) × 3 ÷ 13 + 18 – 5 = ?
What should come in place of the question mark '?' in the following question?
\n13 × 3 of 5 ÷ 26 = ? + 2.5What should come in place of the question mark '?' in the following question?
10% of 450 - (8 × 5) = ? × 5
What will come in the place of the question mark ‘?’ in the following question?
(66)2 - (34)2 + ?2 = (60)2
What will come in the place of the question mark ‘?’ in the following question?
3 × ? ÷ 9 = ∛729 × √121
