Exponential Operations - SAT Math

Although each base is different, we can convert them to a common base of

We know

,

,

and

.

Remember to apply the power rule of exponents.

Therefore we have

.

We can factor out to get

.

When adding exponents, you want to factor out to make solving the question easier.

We can factor out to get

.

Now we can add exponents and therefore our answer is

.

Express as a power of ; that is: .

Then .

Using the properties of exponents, .

Therefore, , so .

(2_n_ * 2) / (2_n_ * 2_n_) simplifies to 2/2_n_ or 21/2_n_.

When dividing exponents with the same base, you subtract the divisor from the dividend, giving 21–n.

When dividing terms with the same base, we can subtract the exponents:

9 – 3 = 6

Start by simplifying the numerator and denominator separately. In the numerator, (c3)2 is equal to c6. In the denominator, c2 * c4 equals c6 as well. Dividing the numerator by the denominator, c6/c6, gives an answer of 1, because the numerator and the denominator are the equivalent.

According to the rules of exponents, ax/ay = ax-y

In this expression, we can follow this rule to simplify x2/x3 and y4/y3

x2–3 = x–1 = 1/x. y4–3 = y1 = y.

Therefore, y/xz2

When dividing, subtract exponents (xa/xb = x(a – b).) Therefore, the quantity in the parenthesis is: x(4 – (–2)) * y(–3 – (–3)) * z(–1 – 5) = x6/z6. Raising this to the 3/2 power results in multiplying the exponents by 3/2: x6 * 3/2/z6 * 3/2 = x9/z9.

If half of the sample decays each week: 1/2 is left after one week, 1/4 is left after two weeks, 1/8 is left after three weeks and 1/16 is left after four weeks (28 days.) That means that 15/16 has decayed. 15/16 x 1010 = 9. 375 x 109

The numerator is simplified to (by adding the exponents), then cube the result. a24/a6 can then be simplified to .

Properties of exponents suggests that when multiplying the same base, add the exponents, when dividing, subtract the exponents on bottom from those on top, and when raising an exponent to another power, multiply the exponents. Remember that (x4/x5) = x–1 = 1/x; Still using order of operations (PEMDAS) we get the following:(x4y2/x5)3= (y2/x)3 = y6/(x3).

x7 / x-3/2 = x7 (x+3/2) based on the fact that division changes the sign of an exponent.

x7 (x+3/2) = x7+3/2 due to the additive property of exponent numbers that are multiplied.

7+3/2= 14/2 + 3/2 = 17/2 so

x7 / x-3/2 = x7+3/2 = x17/2

Since x7 / x-3/2 = xn, xn = x17/2

So n = 17/2

1. According to the rules of exponents, one can add the exponents when adding to variables with the same base. So, x2x4 becomes x6.

2. The rules of exponents also state that if the bases are the same, one can substract the exponents when dividing. So, x6/x becomes x5. Similarly, y/y2 becomes 1/y.

3. When combining these operations, one gets x5/y.

25 = 5 * 5 = 52. Then 54 / 25 = 54 / 52.

Now we can subtract the exponents because the operation is division. 54 / 52 = 54 – 2 = 52 = 25. The answer is therefore 25.

Divide the coefficients and subtract the exponents.

(xy)4 can be rewritten as x4y4 and z0 = 1 because a number to the zero power equals 1. After simplifying, you get 1/y.

The easiest way to solve this is to simplify the fraction as much as possible. We can do this by factoring out the greatest common factor of the numerator and the denominator. In this case, the GCF is .

Now, we can cancel out the from the numerator and denominator and continue simplifying the expression.

The key to this problem is understanding how exponents divide. When two exponents have the same base, then the exponent on the bottom can simply be subtracted from the exponent on top. I.e.:

Keeping this in mind, we simply break the problem down into prime factors, multiply out the exponents, and solve.

When dividing exponents, we need to make sure we have the same base. In this case we do. Then we just subtract the exponents. The answer is .