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.

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.

Divide the coefficients and subtract the exponents.

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.

(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.

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

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

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

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).