Simplify (3x^6b^5)/7*14/(15x^7b^3)
The question asks you to simplify the given algebraic expression. The expression is a fraction involving exponents, and you are required to perform operations consistent with the laws of exponents and arithmetic to reduce the expression to its simplest form. You need to handle multiplication and division of the terms, as well as simplify any like terms by subtracting the exponents when dividing powers with the same base as per the rules of exponents.
$\frac{3 x^{6} b^{5}}{7} \cdot \frac{14}{15 x^{7} b^{3}}$
Answer
Solution:
Step 1
Merge the given expression: $\frac{3x^{6}b^{5} \cdot 14}{7 \cdot 15x^{7}b^{3}}$
Step 2
Eliminate the common factor between 3 and 15.
Step 2.1
Extract the factor 3 from $3x^{6}b^{5} \cdot 14$: $\frac{3(x^{6}b^{5} \cdot 14)}{7(15x^{7}b^{3})}$
Step 2.2
Proceed to reduce common factors.
Step 2.2.1
Extract the factor 3 from $7(15x^{7}b^{3})$: $\frac{3(x^{6}b^{5} \cdot 14)}{3(7(5x^{7}b^{3}))}$
Step 2.2.2
Remove the common factor 3: $\frac{\cancel{3}(x^{6}b^{5} \cdot 14)}{\cancel{3}(7(5x^{7}b^{3}))}$
Step 2.2.3
Reformulate the expression: $\frac{x^{6}b^{5} \cdot 14}{7(5x^{7}b^{3})}$
Step 3
Eliminate the common factor between $x^{6}$ and $x^{7}$.
Step 3.1
Extract the factor $x^{6}$ from $x^{6}b^{5} \cdot 14$: $\frac{x^{6}(b^{5} \cdot 14)}{7(5x^{7}b^{3})}$
Step 3.2
Proceed to reduce common factors.
Step 3.2.1
Extract the factor $x^{6}$ from $7(5x^{7}b^{3})$: $\frac{x^{6}(b^{5} \cdot 14)}{x^{6}(7(5x b^{3}))}$
Step 3.2.2
Remove the common factor $x^{6}$: $\frac{\cancel{x^{6}}(b^{5} \cdot 14)}{\cancel{x^{6}}(7(5x b^{3}))}$
Step 3.2.3
Reformulate the expression: $\frac{b^{5} \cdot 14}{7(5x b^{3})}$
Step 4
Eliminate the common factor between $b^{5}$ and $b^{3}$.
Step 4.1
Extract the factor $b^{3}$ from $b^{5} \cdot 14$: $\frac{b^{3}(b^{2} \cdot 14)}{7(5x b^{3})}$
Step 4.2
Proceed to reduce common factors.
Step 4.2.1
Extract the factor $b^{3}$ from $7(5x b^{3})$: $\frac{b^{3}(b^{2} \cdot 14)}{b^{3}(7(5x))}$
Step 4.2.2
Remove the common factor $b^{3}$: $\frac{\cancel{b^{3}}(b^{2} \cdot 14)}{\cancel{b^{3}}(7(5x))}$
Step 4.2.3
Reformulate the expression: $\frac{b^{2} \cdot 14}{7(5x)}$
Step 5
Eliminate the common factor between 14 and 7.
Step 5.1
Extract the factor 7 from $b^{2} \cdot 14$: $\frac{7(b^{2} \cdot 2)}{7(5x)}$
Step 5.2
Proceed to reduce common factors.
Step 5.2.1
Remove the common factor 7: $\frac{\cancel{7}(b^{2} \cdot 2)}{\cancel{7}(5x)}$
Step 5.2.2
Reformulate the expression: $\frac{b^{2} \cdot 2}{5x}$
Step 6
Rearrange the terms: $\frac{2b^{2}}{5x}$
Knowledge Notes:
The problem involves simplifying a complex rational expression. The process includes combining the terms, factoring out common factors, and canceling them to simplify the expression. Here are the relevant knowledge points:
Combining Terms: When simplifying expressions, it is often useful to combine all terms into a single fraction to see the overall structure and identify common factors.
Factoring: Factoring is the process of breaking down expressions into their constituent parts. This is useful for identifying and canceling out common factors between the numerator and the denominator.
Canceling Common Factors: When a factor appears in both the numerator and the denominator, it can be canceled out. This is based on the property that a number divided by itself equals one.
Exponent Rules: When dealing with exponents, remember that $x^{m}/x^{n} = x^{m-n}$ if $m > n$. This is used to simplify expressions with variables raised to powers.
Rearranging Terms: After canceling, it can be helpful to rearrange terms for clarity or to match a desired format.
Simplification: The goal of simplification is to write the expression in the simplest form, which often means having the fewest terms and lowest exponents possible.
By applying these principles systematically, the initial complex expression can be simplified to a much simpler form.
