Question

1. Simplify using only positive exponents:

(2t)⁻⁶

2. Simplify using only positive exponents:

(w⁻²j⁻⁴)⁻³(j⁷j³)

3. Simplify using only positive exponents:

a²b⁻⁷c⁴
----------
a⁵b³c⁻²

4. Evaluate the expression for m = 2, t = -3, and z = 0.

z⁻ᵗ(mᵗ)ᶻ

5. Use scientific notation to rewrite the number:
a. 0.0002603 in scientific notation
b. 5.38 × 102 in standard notation

Answer 1

1. To simplify this using only positive exponents we are going to use the rule for negative exponents:
`b^(-n) = (1)/(b^(n) )`. Notice that in this case
`b=2t`:

`(2t)^(-c) = (1)/((2t)^(c) )`

2. To simplify this one, we are going to use the rule for negative exponents twice, the product rule:
`(a^(n) )(a^(m) )=a^(n+m)`, and the power rule
`(a^(n)) ^(m) =a^((n)(m))`, so:

`(w^(-2) j ^(-4) ) ^(-3) (j^(7) j ^(3) )=( (1)/(w ^(-2)j ^(-4) ) )^(3) (j ^(7+3) )`

`=(w^2j^4)^3(j ^(10))`

`=(w^6j^(12))(j^(10))`

`=w^6j^(22)`

3. To simplify this one we are going to use the rule for negative exponents, the product rule, and the quotient rule:
`(a^n)/(a^m) =a^(n-m)`, so:

`(a^2b^(-7)c^4)/(a^5b^3c^-2) = (a^2c^4c^2)/(a^5b^3b^7) = (a^(-3)c^6)/(b^(10)) = (c^6)/(a^3b^10)`

4. The first thing we need to is apply the exponents rules; in this case our rule for negative exponents:

`z^(-t)(m^t)^z=( (1)/(z^t) )(m^((t)(z)))= (m^(tz))/(z^t)`
Now can replace our numerical values:

`(2^((-3)(0)))/(0^(-3))`
We have a negative exponent in the denominator, so lets apply oir rule for negative exponents again:

`2^((-3)(0))0^3=2^00^3=(1)(0)=0`

5. Scientific notation is just a way of writing large an small numbers using powers of 10. The exponent of 10 will be the number of places we shift the decimal point to write the number in scientific notation. A positive exponent shows that the decimal point is shifted the right, and a negative one shows that the decimal point is shifted to the left:
a.
`0.0002603=2.603` x
`10^(-4)`
b.
`5.38` x
`10^(2) =538`