MathChapter 1: The Language and Tools of Algebra
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Prompt
What is the base in \(7^4\)?
💡It is the repeated factor.
Answer
\(7\)
Example\(7^4=7\cdot7\cdot7\cdot7\).
Prompt
What is the exponent in \(7^4\)?
💡It counts factors.
Answer
\(4\)
ExampleThere are four factors of \(7\).
Prompt
Write \(5\cdot5\cdot5\) as a power.
💡Base is the factor; exponent is its count.
Answer
\(5^3\)
Prompt
Why is \(2^5\) not \(10\)?
💡Do not multiply base and exponent.
Answer
An exponent means repeated multiplication: \(2^5=32\).
ExampleFive factors of \(2\) produce \(32\).
Prompt
Compare \((-3)^2\) and \(-3^2\).
💡Ask whether the sign belongs to the base.
Answer
They are \(9\) and \(-9\), respectively.
ExampleParentheses change the base.
Prompt
What is the first operation level?
💡Fraction bars and absolute values also group.
Answer
Grouping symbols, from innermost outward.
ExampleIn \(2(5-3)\), evaluate \(5-3\) first.
Prompt
What follows grouping in order of operations?
💡Resolve the complete base first.
Answer
Exponents.
ExampleIn \((2+1)^2\), grouping precedes the square.
Prompt
Which comes first: multiplication or division?
💡Treat them as one level.
Answer
Neither universally; they have equal priority and proceed left to right.
Example\(24/6\times2=8\).
Prompt
Which comes first: addition or subtraction?
💡Treat them as one level.
Answer
Neither universally; they have equal priority and proceed left to right.
Prompt
How should a dense expression be evaluated?
💡This preserves signs.
Answer
Simplify one priority level per written line and copy untouched parts exactly.
ExampleResolve a bracket before its outside multiplier.
Prompt
Evaluate \(3^2+4\).
💡Exponent first.
Answer
\(13\)
Prompt
Evaluate \(20/5\times4\).
💡Left to right at the multiplication/division level.
Answer
\(16\)
Example\(20/5=4\), then \(4\times4=16\).
Prompt
Evaluate \(9-4+2\).
💡Left to right at the addition/subtraction level.
Answer
\(7\)
Example\(9-4=5\), then \(5+2=7\).
Prompt
What is the substitution rule for a negative base?
💡Preserve the sign as part of the value.
Answer
Place the negative value in parentheses before applying an exponent.
ExampleIf \(x=-2\), then \(x^3=(-2)^3\).
Prompt
Evaluate \(|-6|+2^3\).
💡Absolute value groups; powers precede addition.
Answer
\(14\)
Prompt
What is special about \(1^n\) for positive integer \(n\)?
💡Every factor is one.
Answer
It always equals \(1\).
Prompt
What happens when an even exponent is applied to a negative base?
💡Negative factors pair.
Answer
The result is positive when the negative sign is inside the base.
Prompt
What happens when an odd exponent is applied to a negative base?
💡One negative factor remains unpaired.
Answer
The result is negative.
Prompt
How do you evaluate an expression in the exponent?
💡The exponent may itself be an expression.
Answer
Evaluate the exponent's grouping before applying the power.
Example\(2^{3+1}=2^4=16\).
Prompt
SAT check: why inspect every choice after calculating?
💡Distractors often encode order errors.
Answer
To ensure the computed value matches exactly one option and avoid selecting a familiar wrong-order result.
ExampleA division-first error may have its own choice.
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