*I have a take home test for algebra on things that we've done in the class, this is easy work but we did it all so long ago that I can't for the life of me remember how to do it. I'd really appreciate it if someone could help me by tonight. (x stands for multiplication unless noted)*
1. -3 x m x 4 x n x n
2. [ 3a(to the 4th power) t(to the 4th power)] to the 3rd power
3. [2/5] to the 3rd power
4. 2/23 = x/92
5. 10/a = 15/30
6. 2/9 divided by [- 3/27]
Sorry if the way I typed this doesn't make sense. I just don't remember where to start on these problems |

Alright.

1. When you multiply together numbers/variables with the same base, you can simply add their exponents. n can be written as n^1 so

n x n is simply n^2. -3 x 4 = -12. I like to arrange expressions like these in terms of decreasing degree, so I'd write out the answer as -12(n^2)m. Parentheses wouldn't be needed when you write out the answer on paper because the superscript makes it apparent that m is not part of the exponent.

2. When you exponentiate an expression that already has been raised to a power, you multiply the exponents together. Thus, when you have [(3a^4)(t^4)]^3 both the a and t will be raised to the 12th power and the 3 is raised to the third power. Therefore, the simplification would be 27(a^12)t^12. However, though you didn't write it this way, if all of 3a was put to the fourth power (e.g. (3a)^4), the answer would be different so I'm hoping you didn't mistype.

3. When you exponentiate a number, you simply multiply the number by itself for as many times as the value of the exponent. So (2/5)^3 = 2/5 x 2/5 x 2/5 = 8/125. Alternatively you could think about this problem similarly to the previous 2. (2/5)^3 = 2^3/5^3 = 8/125.

4. This is just algebra. What do we need to do to get x by itself? Currently x is being divided by 92 so to get x by itself we must multiply both sides of the equation by 92. In numbers, 92 x 2/23 = x/92 x 92. Simplified, 8 = x.

5. This is essentially the same problem as 4 except that the variable is in the denominator now. So for this one, you could either "flip" both sides of the equation (e.g. a/10 = 30/15, then approach the problem the same way as 4) or cross-multiply (e.g. 10 x 30 = 15a and divide by 15 on both sides to get a by itself). Either way, a = 20.

6. When dividing by fractions, it's often easiest to just multiply by the reciprocal. So 2/9 x -27/3 = 2/9 x -9 = -2.