Simplify 5 x times the fraction 1 over x to the power of negative 7 times x to the power of negative 2. 5x 5x−6 5 5x6

Mathematics

1 answer:

nasty-shy [4]2 years ago

4 0

Answer:

$5x^6$

Explanation:

The expression that you want to simplify is:

$5x\times \frac{1}{x^{-7}}\times x^{-2$

This is the simplification step-by-step:

1. Mulitply the expressions on the numerator among them and the expressions on the denominator among them:

$5x\times \frac{1}{x^{-7}}\times x^{-2}=\frac{5x\cdot x^{-2}}{x^{-7}}$

2. For equal bases that are multiplying, add the exponents:

$\frac{5x\cdot x^{-2}}{x^{-7}}=\frac{5x^{1-2}}{x^{-7}}=\frac{5x^{-1}}{x^{-7}}$

3. Pass the power on the denominator to the numerator by changing its sign (negative exponents become positive when inverted)

$\frac{5x^{-1}}{x^{-7}}=5x^{-1}x^{7}$

4. Simplify adding the exponents with the same base:

$5x^{-1}x^{7}=5x^{-1+7}=5x^6$

And that is the final expression in its most simple form.

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Can anyone explain those marked steps in case (iii)

Tema [17]

$\cos\dfrac{3x}2=\cos\left(\dfrac\pi2+\dfrac x2\right)$
$\implies \dfrac{3x}2=2n\pi+\dfrac\pi2+\dfrac x2$

follows from the fact that the cosine function is $2\pi$ -periodic, which means $\cos x=\cos(2\pi+x)$ . Roughly speaking, this is the same as saying that a point on a circle is the same as the point you get by completing a full revolution around the circle (i.e. add $2\pi$ to the original point's angle with respect to the horizontal axis).

If you make another complete revolution (so we're effectively adding $4\pi$ ) we get the same result: $\cos x=\cos(4\pi+x)$ . This is true for any number of complete revolutions, so that this pattern holds for any even multiple of $\pi$ added to the argument. Therefore $\cos x=\cos(2n\pi+x)$ for any integer $n$ .

Next, because $\cos(-x)=\cos x$ , it follows that $\cos x=\cos(2n\pi-x)$ is also true for any integer $n$ . So we have

$\implies \dfrac{3x}2=2n\pi\pm\left(\dfrac\pi2+\dfrac x2\right)$

The rest follows from considering either case and solving for $x$ .