Power rules, roots, and rational vs. irrational numbers
Extend exponent laws into radicals, rational exponents, and the real number system.
Common CoreN-RN.A.1N-RN.A.2N-RN.B.38.EE.A.18.EE.A.4
8 lessons29 short videos7 key concepts, out loud
9.1
Product, Quotient & Power Rules
Combine powers with the core exponent laws
8.EE.A.1
Key idea
To simplify exponential expressions with the same base, add exponents when multiplying (Product Rule), subtract exponents when dividing (Quotient Rule), and multiply exponents when raising a power to a power (Power Rule).
Learn how to simplify expressions by adding, subtracting, or multiplying exponents when combining powers with matching bases.
Key concept · Lesson 9.1
Keep the base, move the exponents
Listen0:00 / 0:25
Transcript. Every one of these rules needs the same base. Multiplying lines the factors up in a row, so you add the exponents. Dividing cancels them, so you subtract. Raising a power to another power multiplies them. Keep the base itself unchanged, and never add exponents when one power is stacked on another.
9.2
Zero & Negative Exponents
Simplify expressions with zero and negative exponents
8.EE.A.1
Key idea
Any non-zero base raised to the power of zero equals 1, and a negative exponent represents the reciprocal of the base raised to the corresponding positive power.
Learn why any non-zero number raised to the zero power equals one, and master how to rewrite negative exponents as positive fractions.
Key concept · Lesson 9.2
Flip it, never negate it
Listen0:00 / 0:22
Transcript. A negative exponent never makes the answer negative. It means the reciprocal, so move that factor across the fraction bar and the exponent turns positive. Only the part carrying the negative exponent moves; a plain coefficient stays where it is. Any non-zero base to the zero power equals one.
9.3
Rational Exponents & Roots
Connect a one-half power to a square root
N-RN.A.1N-RN.A.2
Key idea
A rational exponent represents both a root and a power, where the denominator of the fractional exponent determines the index of the root and the numerator determines the power of the base.
Learn how to translate expressions between radical form and rational exponent form, and evaluate numbers raised to fractional powers.
Key concept · Lesson 9.3
Bottom is the root, top is the power
Listen0:00 / 0:19
Transcript. A fractional exponent is a root and a power at once. The bottom number tells you which root to take, and the top number is the power. Take the root first so the numbers stay small, then raise the result. Do not treat the fraction as division.
9.4
Simplifying Radicals
Tidy up square roots and rationalize denominators
N-RN.A.2
Key idea
To simplify a square root, factor out the largest perfect square from the radicand, and to rationalize a denominator, multiply both the numerator and denominator by the radical to eliminate the root from the bottom.
Learn how to simplify square roots using perfect square factors and rewrite fractions to remove radicals from the denominator.
Key concept · Lesson 9.4
Largest square out, no square root below
Listen0:00 / 0:19
Transcript. Pull out the largest perfect square you can find, or you will end up simplifying twice. The number that escapes the radical multiplies whatever is already outside; it never goes back in. A root left underneath a fraction is not finished, so multiply top and bottom by that root.
9.5
Rational vs. Irrational Numbers
Sums and products of each kind
N-RN.B.3
Key idea
Rational numbers can be written as a ratio of integers, while irrational numbers cannot; adding or multiplying a rational with an irrational always results in an irrational number, except when multiplying by zero.
Learn how to distinguish between rational and irrational numbers, and predict whether their sums and products will be rational or irrational.
Key concept · Lesson 9.5
Roots can land on either side
Listen0:00 / 0:24
Transcript. A rational number can be written as a ratio of two integers, and an irrational one cannot. Simplify first, because a root of a perfect square is a whole number. Adding or multiplying a rational with an irrational keeps it irrational, unless you multiply by zero. Two matching square roots also multiply back to a whole number.
9.6
Pythagorean Theorem & the Distance Formula
Roots in action on the coordinate plane
8.G.B.8
Key idea
The Pythagorean theorem, a^2 + b^2 = c^2, can be applied to a coordinate plane to derive the distance formula, allowing you to calculate the exact distance between any two points.
Learn how to use the Pythagorean theorem to find missing side lengths and calculate the exact distance between any two points on a coordinate plane.
Key concept · Lesson 9.6
Same triangle, coordinate legs
Listen0:00 / 0:19
Transcript. The distance between two points is the hypotenuse of a right triangle. The horizontal gap is one leg and the vertical gap is the other, so square each gap, add them, then take one square root of the total. Never take the root of each piece separately.
9.7
Scientific Notation Arithmetic
Multiply, divide, add, and subtract in scientific notation
8.EE.A.4
Key idea
To perform arithmetic in scientific notation, multiply or divide the coefficients and apply exponent laws to the powers of ten, or rewrite terms to have matching exponents before adding or subtracting.
Learn how to multiply, divide, add, and subtract numbers in scientific notation without converting them to standard form first.
Key concept · Lesson 9.7
Multiplying splits into two jobs
Listen0:00 / 0:25
Transcript. When you multiply, multiply the front numbers and add the powers of ten. When you divide, subtract them. Adding is different: the powers must match first, so rewrite the smaller term until they do, then add the front numbers. Finish by sliding the decimal so the front number sits between one and ten.
9.8
Review & Assessment
Exponent and radical rules
Review, practice, quiz and test
Key idea
Review exponent and radical rules with targeted practice, a practice test, and a chapter test.
This is one chapter of the climb.
Algebra 1 spans 16 chapters, every lesson taught on video with a key concept you can hear, and Vector on hand to answer questions out loud.
