Exponents in Algebra
If you are like most Canadians, your employer pays you biweekly. Assume you earn as follows:
12.00 × (hours worked during the biweekly pay period)
The quantity “hours worked during the biweekly pay period” is the unknown variable. Recall that the word variable simply represents a quantity that can vary in value. Notice that the expression above appears lengthy when you write out the explanation for the variable. Algebra uses symbols to make such expressions more convenient to manipulate. To shorten the expression, making it easier to read, we can assign a letter or a group of letters to represent the variable. In this case, you might choose to represent “hours worked during the biweekly pay period.” This rewrites the above expression as follows:
or simply
Unfortunately, the word algebra makes many people worry. But remember that algebra is just a set of tools that can help in solving a numerical problem. It is used to demonstrate how the pieces of a puzzle fit together to arrive at a solution.
For example, you have used your algebraic skills if you have ever programmed a formula into Microsoft Excel. You told Excel there was a relationship between cells in your spreadsheet. Perhaps your calculation required cell C3 to be divided by the sum of cells C3, D3 and E3, and then multiplied by cell B7. This is an algebraic expression. Excel then took your algebraic expression and calculated the solution by automatically substituting in the appropriate values from the referenced cells (your variables). See the example below:
C7=CC
E
latex h
latex $12
latex 12h
latex h
latex 12h
latex 12\cdot h
latex 12
latex h
latex =
latex 6x+3y
latex 4x+3
latex x
latex y
latex 6x+3y=4x+3
latex x
latex y
latex x
latex y
latex 6x + 3y
latex 6x
latex 3y
latex 12h
latex 6x + 3y
latex 6x
latex 6
latex x
latex 6x
latex \frac{7x}{3}+4xy^2=x^3-2y
latex \frac{7x}{3}
latex 4xy^2
latex \frac{7x}{3}
latex \frac{7}{3}x
latex \frac{7}{3}
latex x
latex \frac{7}{3}
latex x
latex x^3
latex 2y
latex x^3
latex x
latex x
latex x
latex 1\cdot x^3
latex -2y
latex -2
latex y
latex n
latex 1, 2, 3, \ldots
latex b
latex b^n
latex 2^3 = 8latex 2 × 2 × 2
latex y^ay^b = y^{a+b}
latex \dfrac{y^a}{y^b} = y^{a-b}
latex (x^a)^b = x^{ab}
latex (xy)^a=x^ay^a
latex \left(\dfrac{x}{y}\right)^a=\dfrac{x^a}{y^a}
latex y^0 = 1
latex y^{−a} = \frac{1}{y^a}
latex y^{\frac{a}{b} }= \sqrt[b]{y^a}
latex 1
latex 1
latex 2
latex 2^1
latex 2
latex (yz)^a
latex (y^1z^1)^a
latex (y^{1a}z^{1a})
latex y^az^a
latex 1
latex 7q + 4q + 14qlatex q
latex 25q
latex 7q + 4q + 14q
latex 7qlatex 4q
latex q
latex 7q
latex 4q^2
latex q
latex q^2
latex 1
latex x
latex 1x
latex \frac{x}{4}
latex \frac{{1x}}{4}
latex \frac{1}{4}x
latex 7x^0
latex 7(1)
latex 7
latex \frac{3}{2}x + 4{x^2} – 10x – 2y + \frac{x}{3}
latex 23{g^2} – \frac{{17{g^2}}}{5} + {g^4} + {g^2} – \frac{2}{3}{g^2} – 0.15g + {g^3}
latex x
latex g^2
latex 2(4a + 6b) − (2a − 3b)latex 2(4a + 6b) + (−1)(2a − 3b)
latex −(2a − 3b)
latex −2a + 3b
latex 2(4a + 6b) − 2a + 3b
latex 2 − a + 3blatex \frac{{4a}}{{4a}}
latex \frac{{4a}}{{4a}} = 0
latex \frac{4}{4} = 1
latex \frac{a}{a} = 1
latex \frac{{4a}}{{4a}} = \left( 1 \right)\left( 1 \right) = 1
latex \frac{{{a^1}}}{{{a^1}}} = {a^{1 – 1}} = {a^0} = 1
latex {PV} = \frac{FV}{{1+rt}}latex FV = $5,443.84
latex r = 0.12
latex t = \frac{270}{365}
latex FVlatex F
latex V
latex d_1
latex d_2
latex 2x + 5x
latex 7x
latex x = 2
latex 2x + 5x
latex 2(2) + 5(2) = 14
latex 7(2) = 14
latex 2a − 3a + 4 + 6a − 3
latex 5b(4b + 2)
latex \frac{{6{x^3}+12{x^2}+13.5x}}{{3x}}
latex {(1 + i)^3} \cdot {(1 + i)^{14}}
latex {8^{\frac{2}{3}}}
latex I = Prt
latex P = $2,500, r = 0.06
latex (6r^2+10-6r+4r^2-3)-(-12r-5r^2+2+3r)
latex \left[ \dfrac{5x^9 + 3x^9}{2x} \right]^5
latex \dfrac{t}{2} + 0.75t – {t^3} + \dfrac{{5{t^4}}}{t} – \dfrac{{2\left( {t + {t^3}} \right)}}{4}
latex \dfrac{14\left( 1+i \right)+21\left( 1+i\right)^4 – 35\left( 1+i\right)^7}{7\left(1+i \right)}
latex \dfrac{R}{1+0.08\cdot \frac{183}{365}} + 3R\left( 1 + 0.08 \cdot\frac{52}{365} \right)
latex \left[ \left( \dfrac{2}{5}\right)^2\right]^2
latex PV = \dfrac{FV}{\left( 1+i\right)^N}
latex FV = $3,417.24
latex i = 0.05
latex N = 6
latex FV = $10,000, N = 17
latex i = 0.10
latex \left[\dfrac{10a^2b^3c^4}{5b^3c^4}\right]^2 + 6\left(a^8\right)^{1/2} – \left( 3a^2 + 6\right)\left(3a^2 – 3\right)
latex \dfrac{- \left(5x+4y+3\right)\left(2x-5y\right)-\left(10x-2y\right)\left(2y+3\right)}{5}
latex \dfrac{\left(-3z \right)^3}{\left(3z^2 \right)^2}{\left(2z^3\right)}^{ – 4}
latex PMT = $500latex i = 0.05
latex Δ\% = 0.02
latex CY = 2
latex PY = 4
latex N = 20
latex 50,000\cdot \left(1 + \dfrac{0.10}{12}\right)^{ – 27}
latex 995\left[\dfrac{1-\left(1+0.02\right)^{13}\left(1+\frac{0.09}{4}\right)^{ – 13}}{\frac{0.09}{4}-0.02}\right]$