Question 202433
Let's review the rules of exponents. In these rules, the "variables" stand for any expression (unless there is a note to the contrary). They are numbered so they can be referred to by number, <b>not</b> because any of them are more important or useful than another.)<ol><li>A power of a power: {{{(a^n)^m = a^(n*m)}}}</li><li>Multiplication with the same base: {{{a^n * a^m = a^(n + m)}}} (Pay close attention to the difference between this rule and rule #1!)</li><li>Division with the same base: {{{(a^n)/(a^m) = a^(n - m)}}}</li><li>Negative exponents: {{{a^(-n) = 1/(a^n)}}} or {{{1/(a^(-n)) = a^n}}}. In words, {{{a^(-n)}}} stands for the reciprocal of a^n.</li><li>Zero exponents: {{{a^0 = 1}}} Note: a must <b>not</b> be zero!</li><li>Fractional exponents (where "n" and "m" are positive integers):<ul><li>{{{a^(1/n) =  root(n, a)}}}. So {{{a^(1/2) = sqrt(a)}}} , {{{a^(1/3) = root(3, a)}}} , {{{a^(1/4) = root(4, a)}}} , etc.}}}</li><li>Using the above rule and rule #1 together: {{{a^(n/m) = ( root(m, a))^n = root(m, a^n)}}}</li></ul><li>The pseudo-distributive property. <b>This property is not the distributive property</b> but it does look a little like the distributive property: {{{(a*b)^n = a^n*b^n}}}</li><li>Exponents apply only to what is <b>immediately</b> in front of them! If the symbol immediately in front of an exponent is a grouping symbol, then the exponent is applied to the entire expression in the grouping symbol. Examples:<ul><li>{{{4^2 = 4*4}}}</li><li>{{{-4^2 = -(4*4) = -16}}}</li><li>{{{(-4)^2 = (-4)*(-4) = 16}}}. Note the difference between this example and the previous one. Here a ")" is right in front of the exponent. So the exponent applies to everything in the parentheses. In the previous example, a "4" is right in front of the exponent. So the exponent applies only to the 4 (and not the "-"!)</li><li>{{{x + 3^2 = x + 3*3 = x + 9}}}</li><li>{{{(x + 3)^2 = (x + 3)*(x + 3) = x^2 +6x +9}}}</li></ul></li></ol>
Now let's try to solve your problems.
1) Use the rules of exponents to evaluate or simplify. Write without negative exponents.
3*4^0 = ____
According to PEMDAS (the order of operations) we need to simplify the exponent before doing the multiplication. So we need to figure out 4^0 first. Using rule #5 we find that 4^0 = 1. Substituting this in we get
3*1 = ____
which, of course, is 3.<br>

2) Use the rules of exponents to evaluate or simplify. Write without negative exponents.
1 / 4^-2 = ____
Using the 2nd variation of rule #4:
{{{1/(4^(-2)) = 4^2 = 16}}}<br>

3) Use the rules of exponents to evaluate or simplify. Write without negative exponents.
(1/4)^-1/2 = _____
Using rule #1, in "reverse", we can rewrite this as:
{{{((1/4)^(-1))^(1/2)}}}
Then, using rule #4, the expression in the parentheses can be simplified: {{{(1/4)^(-1) = 4^1 = 4}}}. Substituting we get:
{{{(4)^(1/2)}}} which, according to rule #6, is {{{sqrt(4) = 2}}}<br>

4) Use the rules of exponents to evaluate or simplify. Write without negative exponents.
(bm)^-3 = ____
Using rule #1 again we can rewrite this as:
{{{((bm)^3)^(-1)}}}
Using rule 7:
{{{(b^3*m^3)^(-1)}}}
Using rule #4:
{{{1/(b^3*m^3)}}}<br>

5) Use the rules of exponents to evaluate or simplify. Write without negative exponents.
y^-5 / y^-2 = ____
Using rule #3
{{{y^(-5)/(y^(-2)) = y^((-5 - (-2))) = y^(-3)}}}
Using rule #4 we get
{{{y^(-3) = 1/y^3}}}<br>

6) Use the rules of exponents to evaluate or simplify. Write without negative exponents.
(-243)^3/5 = ____
Using the second variation of rule #6:
{{{(-243)^(3/5) = (root(5, -243))^3}}}
Since {{{(-3)^5 = -243}}} {{{root(5, -243) = -3}}}. Substituting into the above we get
{{{(-3)^3 = -27}}}<br>

7) Use the rules of exponents to evaluate or simplify. Write without negative exponents.
36^1/2 = ____
Using rule #6
{{{36^(1/2) = sqrt(36) = 6}}}