Question

If a is equals to 2 power m and b is equals to 2 power m + 1 show that 8 a power 3 by b power 2 is equal 2 m + 1

Answer 1

Answer:

If n is a positive integer and x is any real number, then xn corresponds to repeated multiplication

xn=x×x×⋯×xn times.

We can call this “x raised to the power of n,” “x to the power of n,” or simply “x to the n.” Here, x is the base and n is the exponent or the power.

From this definition, we can deduce some basic rules that exponentiation must follow as well as some hand special cases that follow from the rules. In the process, we'll define exponentials xa for exponents a that aren't positive integers.

Rule or special case Formula Example

Product xaxb=xa+b 2223=25=32

Quotient xaxb=xa−b 2322=21=2

Power of power (xa)b=xab (23)2=26=64

Power of a product (xy)a=xaya 36=62=(2⋅3)2=22⋅32=4⋅9=36

Power of one x1=x 21=2

Power of zero x0=1 20=1

Power of negative one x−1=1x 2−1=12

Change sign of exponents x−a=1xa 2−3=123=18

Fractional exponents xm/n=xm−−−√n=(x√n)m 43/2=(4√)3=23=8

The rules

Product of exponentials with same base

If we take the product of two exponentials with the same base, we simply add the exponents:

xaxb=xa+b.(1)

To see this rule, we just expand out what the exponents mean. Let's start out with a couple simple examples.

3432=(3×3×3×3)×(3×3)=3×3×3×3×3×3=36

y2y3=(y×y)×(y×y×y)=y×y×y×y×y=y5

The general case works the same way. We just need to keep track of the number of factors we have.

xaxb=x×⋯×xa times×x×⋯×xb times=x×⋯×xa+b times=xa+b

Quotient of exponentials with same base

If we take the quotient of two exponentials with the same base, we simply subtract the exponents:

xaxb=xa−b(2)

This rule results from canceling common factors in the numerator and denominator. For example:

y5y3=y×y×y×y×yy×y×y=(y×y)×(y×y×y)y×y×y=y×y=y2.

To show this in general, we look at two different cases. If we imagine that a>b, then this rule follows from canceling the common b factors of x that occur in both the numerator and denominator. We are left with just b−a factors of x in the numerator.

xaxb=x×⋯×xa timesx×⋯×xb times=x×⋯×xa−b times×x×⋯×xb timesx×⋯×xb times=x×⋯×xa−b times=xa−b

If a<b, then what happens? We cancel all the x's from the numerator and are left with b−a of them in the denominator.

xaxb=x×⋯×xa timesx×⋯×xb times=x×⋯×xa timesx×⋯×xb−a times×x×⋯×xa times=1x×⋯×xb−a times

To make the above rule work for this case, we must define a negative exponent to mean a power in the denominator. If n is a positive integer, we define

x−n=1x×x×⋯×xn times.(3)

Then the rule for the quotient of exponentials works even if a<b:

xaxb=x×⋯×xa timesx×⋯×xb times=1x×⋯×xb−a times=xa−b.

When b>a, the exponent a−b is a negative number. Since formula (2) is the same no matter the relationship between a and b, we don't need to worry about it and can just subtract the exponents.

(xa)b=xab.(4)

We can see this result by writing it as a product where the xa is repeated b times:

(xa)b=xa×xa×⋯×xab times.

Next we apply rule (1) for the product of exponentials with the same base. We use this rule b times to conclude that

(xa)b=xa×xa×⋯×xab times=xa+a+⋯+ab times=xab.

In the last step, we had to remember that multiplication can be defined as repeated addition.

Power of a product

If we take the power of a product, we can distribute the exponent over the different factors:

(xy)a=xaya.(5)

We can show this rule in the same way as we show that you can distribute multiplication over addition. One way to show this distributive law for multiplication is is to remember that multiplication is defined as repeated addition:

(x+y)a=(x+y)+(x+y)+⋯+(x+y)a times=x+x+⋯+xa times+y+y+⋯+ya times=xa+ya.

In the same way, we can show the distributive law for exponentiation:

(xy)a=(xy)×(xy)×⋯×(xy)a times=x×x×⋯×xa times×y×y×⋯×ya times=xaya.

This rule also works for quotients

(xy)a=xaya,

but it does NOT work for sums. For example,

(3+5)2=82=64,

but this is NOT equal to

32+52=9+25=34.

Special cases

The following are special cases that follow from the rules.

The power of one

The simplest special case is that raising any number to the power of 1 doesn't do anything:

x1=x.(6)

The power of zero

As long as x isn't zero, raising it to the power of zero must be 1:

x0=1.

We can see this, for example, from the quotient rule, as

1=xaxa=xa−a=x0.

The expression 00 is indeterminate. You can see that it must be indeterminate, because you can come up with good reasons for it to be two different values.

Answer 2

Answer:

Step-by-step explanation: