Explain how to multiply decimals by powers of 10. Use 8.653 x 10000 as an example, to show how the multiplication is done.

Mathematics

1 answer:

Lera25 [3.4K]3 years ago

4 0

Answer:

We move the decimal point to the right by a number of digit places equal to the power of 10.

Step-by-step explanation:

If we want to multiply a decimal by a power of 10, we move the decimal point to the right by a number of digit places which is the power of the number ten or the number of zeros which are present in the multiplying number 10. If we do not have a digit after the last digit, we put zeros to complete the number of digits until, we have the number of digit places added after the decimal point which are equal to the power of 10.

Using the example given above

8.653 x 10000, we have four zeros. So, in index form it is

8.653 x 10⁴

Since we have the number 10 raised to the power of four or having 4 zeros, we move the decimal point 4 places to the right.

So, 8.653 x 10000 = 8.653 x 10⁴ = 86530

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Recall the binomial theorem.

$(a+b)^n = \displaystyle \sum_{k=0}^n \binom nk a^{n-k} b^k$

1. The binomial expansion of $\left(1+\frac x3\right)^7$ is

$\left(1 + \dfrac x3\right)^7 = \displaystyle\sum_{k=0}^7 \binom 7k 1^{7-k} \left(\frac x3\right)^k = \sum_{k=0}^7 \binom 7k \frac{x^k}{3^k}$

Observe that

$k = 1 \implies \dbinom 71 \left(\dfrac x3\right)^1 = \dfrac73 x$

$k = 2 \implies \dbinom 72 \left(\dfrac x3\right)^2 = \dfrac73 x^2$

When we multiply these by $8-9x$ ,

• $8$ and $\frac73 x^2$ combine to make $\frac{56}3 x^2$

• $-9x$ and $\frac73 x$ combine to make $-\frac{63}3 x^2 = -21x^2$

and the sum of these terms is

$\dfrac{56}3 x^2 - 21x^2 = \boxed{-\dfrac73 x^2}$

2. The binomial expansion is

$\left(2a - \dfrac b2\right)^8 = \displaystyle \sum_{k=0}^8 \binom 8k (2a)^{8-k} \left(-\frac b2\right)^k = \sum_{k=0}^8 \binom 8k 2^{8-2k} a^{8-k} b^k$