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Chapter 2: Problem 1

Simplify each of the following: (a) \(2 a b-4 a c+b a-2 c b+3 b a\) (b) \(3 x^{2} y z-z x^{2} y+4 y x z^{2}-2 x^{2} z y+3 z^{2} y x-3 z y^{2}\) (c) \(c^{p} \times c^{-q} \div c^{-2}\) (d) \(\frac{\left(x^{\frac{1}{2}}\right)^{-\frac{2}{3}} \div\left(y^{\frac{3}{4}}\right)^{2} \times\left(x^{\frac{3}{5}}\right)^{-\frac{5}{3}}}{\left(x^{\frac{1}{4}}\right)^{-1} \times\left(\frac{1}{3}\right)^{6}}\)

Short Answer

Expert verified
(a) 6ab - 4ac - 2bc, (b) 7xyz^2 - 3y^2z, (c) c^{p-q+2}, (d) x^{-3/4} y^{-3/2} / 729.

Step by step solution

01

Simplify expression (a)

To simplify expression (a), we combine like terms. The expression is \(2ab - 4ac + ba - 2cb + 3ba\). The like terms are the terms involving \(ab\) which are \(2ab\), \(ba\), and \(3ba\). First, note that \(ba = ab\), so we have: \(2ab + ab + 3ab - 4ac - 2cb = (2+1+3)ab - 4ac - 2cb\). This simplifies to: \(6ab - 4ac - 2bc\). Therefore, the simplified expression is \(6ab - 4ac - 2bc\).
02

Simplify expression (b)

For this expression, group and combine the like terms. The expression is \(3x^2yz - zx^2y + 4yxz^2 - 2x^2zy + 3z^2yx - 3zy^2\). First, notice that \(x^2yz\) terms can be rewritten as: \((3x^2yz - x^2yz - 2x^2yz) + (4yzx^2 + 3yz^2x) - 3zy^2\). The like terms \(3x^2yz - x^2yz - 2x^2yz\) give: \(0x^2yz\), and the remaining are: \(4z^2xy + 3z^2yx - 3zy^2\). Since some terms cancel, the simplified expression is \(7xyz^2 - 3y^2z\).
03

Simplify expression (c) using exponent rules

Apply the rules of exponents to simplify \(c^p \times c^{-q} \div c^{-2}\). Combining the terms using the rule \(c^a \times c^b = c^{a+b}\) and \(c^a \div c^b = c^{a-b}\), we get: \(c^{p - q} \times c^{2} = c^{p - q + 2}\). Thus, the expression simplifies to \(c^{p - q + 2}\).
04

Simplify expression (d) using exponent rules

Parse the expression carefully. The expression is: \[ \frac{\left(x^{\frac{1}{2}}\right)^{-\frac{2}{3}} \div \left(y^{\frac{3}{4}}\right)^2 \times \left(x^{\frac{3}{5}}\right)^{-\frac{5}{3}}}{\left(x^{\frac{1}{4}}\right)^{-1} \times \left(\frac{1}{3}\right)^{6}} \]. Negate and simplify each power: - \(\left(x^{\frac{1}{2}}\right)^{-\frac{2}{3}}\) becomes \(x^{-\frac{1}{3}}\), - \(\left(y^{\frac{3}{4}}\right)^2\) becomes \(y^{\frac{3}{2}}\), - \(\left(x^{\frac{3}{5}}\right)^{-\frac{5}{3}}\) becomes \(x^{-1}\), - \(\left(x^{\frac{1}{4}}\right)^{-1}\) becomes \(x^{-\frac{1}{4}}\), - \(\left(\frac{1}{3}\right)^6\) stays as it is. The key is making the fraction a multiplication trick: multiply numerators and denominators accordingly. Multiply them stepwise, with results: \[ x^{-\frac{1}{3} - 1} y^{-\frac{3}{2}} \cdot x^{\frac{1}{4}} = x^{-\frac{3}{12}} y^{-\frac{3}{2}} \frac{1}{729} \]. Overall, the expression, after simplification, equals \[ x^{-3/4} y^{-3/2} (1/729) \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Like Terms
In algebra, like terms are terms that have the same variables raised to the same power. The coefficients of these terms can be different, but the variables and their exponents must match. This concept helps simplify expressions by combining these identical parts.

For example, in the expression \[2ab + ba + 3ba - 4ac - 2cb,\]we notice that terms like \(ab\) are present multiple times albeit in slightly different forms like \(ba\) and \(3ba\). Since \(ab\) and \(ba\) are similar, these can be added together.
  • Combine \(2ab\), \(ba\) (which is \(1ab\)), and \(3ba\) (or \(3ab\)) to get \((2+1+3)ab = 6ab\).
  • The terms \(-4ac\) and \(-2cb\) don't have like pairs, so they stay as they are.
Thus, the expression simplifies beautifully into \[6ab - 4ac - 2cb.\]
Rules of Exponents
The rules of exponents are essential when simplifying expressions that involve powers. They help us manipulate expressions more smoothly, allowing us to combine or reduce terms.

Key rules include:
  • When multiplying like bases, add their exponents: \(a^m \times a^n = a^{m+n}\).
  • When dividing like bases, subtract their exponents: \(a^m \div a^n = a^{m-n}\).
  • A power raised to another power requires multiplying the exponents: \((a^m)^n = a^{m \times n}\).
For example, simplifying \[c^p \times c^{-q} \div c^{-2}\]and using the exponent rules gives \[(c^{p-q}) \times c^{2} = c^{p-q+2}.\]These rules simplify complex expressions into more manageable forms.
Fraction Simplification
Fraction simplification involves reducing the complexity of fractions. This can be achieved by cancelling common factors or simplifying denominators and numerators separately.

When dealing with fractions that have variables raised to powers, apply the rules of exponents. For instance, \[\frac{(x^{\frac{1}{2}})^{-\frac{2}{3}} \div (y^{\frac{3}{4}})^2 \times (x^{\frac{3}{5}})^{-\frac{5}{3}}}{(x^{\frac{1}{4}})^{-1} \times (\frac{1}{3})^6}\]requires
  • Careful handling of negative exponents and fractions.
  • Rewriting each power so they're easily simplified.
The end result involves multiplying and simplifying step-by-step until the fraction becomes its smallest form. The crucial part is to keep track of changes in exponents and coefficients.
Negative Exponents
Negative exponents are another essential concept in algebraic simplification. They indicate the reciprocal of the base raised to the corresponding positive exponent. Understanding negative exponents means you can rewrite them to simplify expressions with ease.

For instance, the expression \[(x^{\frac{1}{2}})^{-\frac{2}{3}}\]translates into \[x^{-(\frac{2}{3} \times \frac{1}{2})} = x^{-\frac{1}{3}}.\]Similarly, handling negative exponents in division or multiplication requires flipping the terms or reversing the operation.
  • For any base \(a\), remember: \(a^{-n} = \frac{1}{a^n}\).
  • In operations, convert negative exponents to positivities to observe clear simplifications.
By managing negative exponents correctly, you can achieve a more straightforward, final expression with positive exponents only. This not only makes the expression easier to comprehend but also useful for further mathematical operations.

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Most popular questions from this chapter

Perform the following multiplications and simplify your results: (a) \((3 x-1)\left(x^{2}-x-1\right)\) (b) \(\left(a^{2}+2 a+2\right)\left(3 a^{2}+4 a+4\right)\)

Factorize the following: (a) \(36 x^{3} y^{2}-8 x^{2} y\) (b) \(x^{3}+3 x^{2} y+2 x y^{2}+6 y^{3}\) (c) \(4 x^{2}+12 x+9\) (d) \((3 x+4 y)^{2}-(2 x-y)^{2}\) (e) \(x^{2}+10 x+24\) (f) \(x^{2}-10 x+16\) (g) \(x^{2}-5 x-36\) (h) \(6 x^{2}+5 x-6\)

Express the following without logarithms: (a) \(\log x=\log P+2 \log Q-\log K-3\) (b) \(\log R=1+\frac{1}{3} \log M+3 \log S\) (c) \(\ln P=\frac{1}{2} \ln (Q+1)-3 \ln R+2\)

Express in log form: (a) \(f=\frac{1}{\pi d \sqrt{L C}}\) (b) \(K=\frac{a^{3} \times \sqrt{b}}{c^{\frac{1}{6}} d^{\frac{2}{5}}}\)

Evaluate by calculator or by change of base where necessary: (a) \(\log 0.0270\) (b) \(\ln 47.89\) (c) \(\log _{7} 126 \cdot 4\)
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