Question

Challenge Problems. $$\left(72 p^{6} q^{7} \div 9 p^{4} q\right) \div 8 p q^{6}$$

Short answer

\(p\)

Step-by-step solution

Simplify the Numerator

First, simplify the expression in the numerator by dividing each term by the respective terms in the denominator. In the numerator we have: \(72 p^{6} q^{7}\). Divide each term by the corresponding terms in the denominator: \(9 p^{4} q\). This gives us: \(\frac{72}{9} = 8\), for the coefficients, \(p^{6} \/ p^{4} = p^{2}\) and \(q^{7} \/ q = q^{6}\). The resulting simplified numerator is \(8 p^{2} q^{6}\).

Divide by the Remaining Denominator

Now divide the simplified numerator \(8 p^{2} q^{6}\) by the remaining denominator \(8 p q^{6}\). The coefficients \(8\/8 = 1\). For the variables, \(p^{2} \/ p = p^{1} = p\) and \(q^{6} \/ q^{6} = q^{0} = 1\), since any number to the power of zero is 1. Thus, the entire expression simplifies to \(p\).

Key concepts

Polynomial Division

Polynomial division, much like numerical division, involves dividing one polynomial by another. It's a process used in algebra to simplify expressions or to rewrite them in a different form. The division of polynomials can be a bit more complex than numerical division since it involves variables and their exponents.

When dividing polynomials, one polynomial (the dividend) is divided by another polynomial (the divisor). This process often involves techniques such as long division or synthetic division, especially when handling higher-degree polynomials. However, for simpler cases like in the exercise, we divide the coefficients (numeric parts) and subtract the exponents of like bases. It's key to remember that when we divide powers with the same base, we subtract the exponents due to the exponent rules. For instance, dividing \(p^6\) by \(p^4\) results in \(p^{6-4}=p^2\).

The challenge here is also to be attentive to the variables involved and to apply the basic operation correctly to both the numerical coefficients and the variable parts of the polynomials.

Exponent Rules

Exponent rules, or laws of exponents, are essential for simplifying algebraic expressions involving powers. There are a few key rules that we frequently use:

• The Product of Powers rule states that when multiplying two powers that have the same base, you add the exponents (e.g., \(x^m \times x^n = x^{m+n}\)).
• The Quotient of Powers rule dictates that when dividing two powers with the same base, you subtract the exponents (e.g., \(x^m \/ x^n = x^{m-n}\)).
• The Power of a Power rule tells us that when taking an exponent to another exponent, you multiply the exponents (e.g., \( (x^m)^n = x^{m \cdot n}\)).
• Lastly, the Zero Exponent rule states that any non-zero number raised to the power of zero equals one (e.g., \(x^0 = 1\)).

Applying these rules correctly can greatly simplify the process of solving algebraic problems. In our given exercise, the Quotient of Powers and the Zero Exponent rules are particularly important to arrive at the simplified form of \(p\).

Algebraic Fractions

Algebraic fractions, also known as rational expressions, are fractions where the numerator and/or the denominator contain algebraic expressions. Simplifying these fractions involves factoring, reducing common terms, and following the basic rules of arithmetic operations applied to polynomials.

When simplifying algebraic fractions, the goal is to make the expression as straightforward as possible. Similar to numerical fractions, we look for common factors in the numerator and the denominator that we can divide out. We must be careful, however, to only reduce terms that are common factors; we cannot cancel terms that are added or subtracted within the numerator or denominator. In some cases, we might need to factor polynomials further to find common terms to cancel out.

For our exemplified exercise, the algebraic fractions become much simpler since we have direct terms to divide in the numerator and the denominator. The process is more straightforward and doesn't require factoring as much as reducing terms by division, keeping in mind the correct application of exponent rules, leading to the final answer of \(p\).