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3 years ago

What does 12−3/4(d+16)=−5 equal solve for d

1 answer:

3 0

$12 - \frac{3}{4} (d + 16) = - 5 \\ \\12 - \frac{3}{4} d - 12 = - 5 \\ \\ - \frac{3}{4} d = - 5 \\ \\ - 3d = - 20 \\ \\ d = \frac{20}{3}$

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Can you see an easy way to work out the value of <img src="https://tex.z-dn.net/?f=%5Csqrt%7B%201%5E3%20%2B%202%5E3%20%2B%203%5E

Lina20 [59]

We have the sum of cubes identity

$a^3 + b^3 = (a + b) (a^2 - ab + b^2)$

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and

$2^3 + 3^3 = (2 + 3) (2^2 - 6 + 3^2) = 5 \times 7$

Then

$\sqrt{1^3+2^3+3^3+4^3+5^3} = \sqrt{5\times13 + 5\times7 + 5\times5^2} = \sqrt{5 \times (20 + 25)} \\\\ ~~~~~~~~ = \sqrt{5^2 \times (4 + 5)} = \sqrt{5^2\times9} = \sqrt{5^2\times3^2} = 5\times3 = \boxed{15}$

Alternatively, we have the well-known sum of cubes formula

$\displaystyle \sum_{i=1}^n i^3 = \frac{n^2(n+1)^2}4$

The sum under the square root is this sum with $n=5$ . Then

$1^3+2^3+3^3+4^3+5^3 = \dfrac{5^2\times6^2}4 = 225 = 15^2$

and so the square root again reduces to 15.

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