Simplify (39t^4uv^-2)/(13t^-3u^7)

Solution:

Step 1:

Apply the negative exponent rule $a^{-n}=\frac{1}{a^n}$ to rewrite $v^{-2}$ in the denominator.

$$\frac{39t^{4}u}{13t^{-3}{u}^7{v}^2}$$

Step 2:

Similarly, apply the negative exponent rule to move $t^{-3}$ to the numerator.

$$\frac{39t^{4}u{t}^3}{13u^7{v}^2}$$

Step 3:

Combine like terms by adding exponents of $t$.

Step 3.1:

Rewrite the expression with $t^3$ next to $t^4$.

$$\frac{39(t^3{t}^4)u}{13u^7{v}^2}$$

Step 3.2:

Use the exponent addition rule $a^m\cdot a^n=a^{m+n}$.

$$\frac{39t^{3+4}u}{13u^7{v}^2}$$

Step 3.3:

Perform the addition of exponents.

$$\frac{39t^{7}u}{13u^7{v}^2}$$

Step 4:

Simplify by canceling out common factors between the numerator and the denominator.

Step 4.1:

Extract the common factor of 13 from the numerator.

$$\frac{13(3t^{7}u)}{13u^7{v}^2}$$

Step 4.2:

Proceed to cancel out the common factors.

Step 4.2.1:

Factor out 13 from the denominator.

$$\frac{13(3t^{7}u)}{13(u^7{v}^2)}$$

Step 4.2.2:

Eliminate the common factor of 13.

$$\frac{\cancel{13}(3t^{7}u)}{\cancel{13}(u^7{v}^2)}$$

Step 4.2.3:

Rewrite the simplified expression.

$$\frac{3t^{7}u}{u^7{v}^2}$$

Step 5:

Further reduce the expression by canceling out the common $u$ terms.

Step 5.1:

Factor $u$ from the numerator.

$$\frac{u(3t^7)}{u^7{v}^2}$$

Step 5.2:

Cancel out the common $u$ factors.

Step 5.2.1:

Factor $u$ from the denominator.

$$\frac{u(3t^7)}{u(u^6{v}^2)}$$

Step 5.2.2:

Cancel the common $u$ factor.

$$\frac{\cancel{u}(3t^7)}{\cancel{u}(u^6{v}^2)}$$

Step 5.2.3:

Write down the final simplified expression.

$$\frac{3t^7}{u^6{v}^2}$$

Knowledge Notes:

1. Negative Exponent Rule: The negative exponent rule states that $a^{-n}=\frac{1}{a^n}$ and $\frac{1}{a^{-n}}=a^n$. This rule is used to rewrite expressions with negative exponents to their reciprocal forms with positive exponents.

2. Combining Like Terms with Exponents: When multiplying terms with the same base, you can add the exponents together. This is known as the power rule, which states that $a^m\cdot a^n=a^{m+n}$.

3. Simplifying Fractions: When simplifying fractions, any common factors in the numerator and the denominator can be canceled out. This includes numbers as well as variables with exponents, provided that the base and exponent are identical.

4. Factoring: Factoring involves rewriting an expression as a product of its factors. This can help in identifying and canceling out common factors between the numerator and the denominator of a fraction.

5. Final Expression: After simplifying and canceling out common factors, the final expression should be written with positive exponents and in its simplest form.