Help with simplifying?

Question

How do I simplify the following expressions:

1. 5x^2 * 2x^5
2. (-2c^3)^2
3. t^3 * t^(n-3)
4. (x^m)^n * (x^n)^(n-m)

Showing all the steps will be very helpful, and thank you in advance for the help.

Answer 1

1) factoring the x: (5*2)(x^(2+5))=10X^7
2) (-2^2)c^(2*3) = 4c^6
3) t^(n-3+3) = t^n
4)x ^ (m*n) * x ^ (n(n-m) = (x^mn) (x^ ((n^2)-nm))

Answer 2

1. Exponents rule:
when multiplying powers of the same base (in this case, x), you add the exponents.

x^2 = x*x
x^5 = x*x*x*x*x
therefore
(x^2)*(x^5) = (x*x)*(x*x*x*x*x)
=x*x*x*x*x*x*x = x^7

5x^2*2x^5 = 5*(x^2)*2*(x^5) =
5*2*(x^2)*(x^5) = 10*(x^7) = 10x^7

2. Using the same rule, we find that raising an exponent to a power is the same as multiplying the exponent by the power:
(c^3)^2 = (c^3)*(c^3) = c^6 (using rule from number 1)
same as
(c^3)^2 = c^(3*2) = c^6

(-2c^3)^2 = (-2)^2 * (c^3)^2 = (+4)*(c^6) = 4c^6

3. Same as rule number 1:
t^3 * t^(n-3) = t^[3+(n-3)] = t^(3+n-3) = t^n

4. Combine rules 2 and 1:
First, take each factor separately:

(x^m)^n = x^(m*n) = x^mn

(x^n)^(n-m) = x^[n*(n-m)] = x^(n^2 -m)

therefore:
(x^m)^n * (x^n)^(n-m) = x^mn * x^(n^2 -m) = x^[mn+(n^2 - m)] =
x^(n^2 +mn - m)

Answer 3

When multiplying these expressions you can bring the coefficients to the front an mult, then if the bases are the same, add the exponenets. (see laws of exponents in your textbook)

(5x^2)(2x^5) = 10x^7

power of a power rule

2) (-2c^3)^2 = 4c^6 notice it's positive

3) t^3[t^(n-3)] law of exponents. When the bases are the same you add the exponents.

= t^n b/c 3 + (n-3) = n

4) (x^m)^n * (x^n)^(n-m) power rule first

=( x^mn) [x^(n^2-mn) bases are now the same so you can add the exponents

= x ^(n^2) b/c the mn + (-mn) cancel out.

Answer 4

Problem 1:

a) Add the exponents because the terms are being multiplied (subtract them for division).

b) Multiply the coefficients.

c) 10x^7


Problem 2:

a) Multiply the exponents.

b) Square the coefficient.

c) 4c^6


Problem 3:

a) Add the exponents because the terms are being multiplied.

b) Multiply the coefficients, which are in this case both 1.

c) t^n


Problem 4:

Does not simplify.

Answer 5

1. = 5*x*x*2*x*x*x*x*x = 10x^7
2. = (-2)^2 * (c^3)^2 = 4c^6
3. = t ^(3+n-3) = t^n
4. = x^(mn+n^2-mn) = x^(n^2)