Key Concepts
Square Root Notation mm is read ‘the square root of m’ If n2 = m, then n=m,n=m, for n≥0.n≥0. The square root of m, m,m, is a positive number whose square is m.
nth Root of a Number If bn=a,bn=a, then b is an nth root of a. The principal nth root of a is written an.an. n is called the index of the radical.
Properties of anan When n is an even number and a≥0,a≥0, then anan is a real number a<0,a<0, then anan is not a real number When n is an odd number, anan is a real number for all values of a.
Simplifying Odd and Even Roots For any integer n≥2,n≥2, when n is odd ann=aann=a when n is even ann=|a|ann=|a| We must use the absolute value signs when we take an even root of an expression with a variable in the radical.
Simplified Radical Expression For real numbers a, m and n≥2n≥2 anan is considered simplified if a has no factors of mnmn
Product Property of nth Roots For any real numbers, anan and bn,bn, and for any integer n≥2n≥2 abn=an·bnabn=an·bn and an·bn=abnan·bn=abn
How to simplify a radical expression using the Product Property Step 1. Find the largest factor in the radicand that is a perfect power of the index.Rewrite the radicand as a product of two factors, using that factor. Step 2. Use the product rule to rewrite the radical as the product of two radicals. Step 3. Simplify the root of the perfect power.
Quotient Property of Radical Expressions If anan and bnbn are real numbers, b≠0,b≠0, and for any integer n≥2n≥2 then, abn=anbnabn=anbn and anbn=abnanbn=abn
How to simplify a radical expression using the Quotient Property. Step 1. Simplify the fraction in the radicand, if possible. Step 2. Use the Quotient Property to rewrite the radical as the quotient of two radicals. Step 3. Simplify the radicals in the numerator and the denominator.
Rational Exponent a1na1n If anan is a real number and n≥2,n≥2, then a1n=an.a1n=an.
Rational Exponent amnamn For any positive integers m and n, amn=(an)mamn=(an)m and amn=amnamn=amn
Properties of Exponents If a, b are real numbers and m, n are rational numbers, then Product Property am·an=am+nam·an=am+n Power Property (am)n=am·n(am)n=am·n Product to a Power (ab)m=ambm(ab)m=ambm Quotient Property aman=am−n,a≠0aman=am−n,a≠0 Zero Exponent Definition a0=1,a0=1, a≠0a≠0 Quotient to a Power Property (ab)m=ambm,b≠0(ab)m=ambm,b≠0 Negative Exponent Property a−n=1an,a≠0a−n=1an,a≠0
Product Property of Roots For any real numbers, anan and bn,bn, and for any integer n≥2n≥2 abn=an·bnabn=an·bn and an·bn=abnan·bn=abn
Special Products Binomial SquaresProduct of Conjugates(a+b)2=a2+2ab+b2(a+b)(a−b)=a2−b2(a−b)2=a2−2ab+b2Binomial SquaresProduct of Conjugates(a+b)2=a2+2ab+b2(a+b)(a−b)=a2−b2(a−b)2=a2−2ab+b2
Quotient Property of Radical Expressions If anan and bnbn are real numbers, b≠0,b≠0, and for any integer n≥2n≥2 then, abn=anbnabn=anbn and anbn=abnanbn=abn
Simplified Radical Expressions A radical expression is considered simplified if there are: no factors in the radicand that have perfect powers of the index no fractions in the radicand no radicals in the denominator of a fraction
Binomial Squares (a+b)2=a2+2ab+b2(a−b)2=a2−2ab+b2(a+b)2=a2+2ab+b2(a−b)2=a2−2ab+b2
Solve a Radical Equation Step 1. Isolate one of the radical terms on one side of the equation. Step 2. Raise both sides of the equation to the power of the index. Step 3. Are there any more radicals? If yes, repeat Step 1 and Step 2 again. If no, solve the new equation. Step 4. Check the answer in the original equation.
Problem Solving Strategy for Applications with Formulas Step 1. Read the problem and make sure all the words and ideas are understood. When appropriate, draw a figure and label it with the given information. Step 2. Identify what we are looking for. Step 3. Name what we are looking for by choosing a variable to represent it. Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.
Falling Objects On Earth, if an object is dropped from a height of h feet, the time in seconds it will take to reach the ground is found by using the formula t=h4.t=h4.
Skid Marks and Speed of a Car If the length of the skid marks is d feet, then the speed, s, of the car before the brakes were applied can be found by using the formula s=24d.s=24d.
Properties of anan When n is an even number and: a≥0,a≥0, then anan is a real number. a<0,a<0, then anan is not a real number. When n is an odd number, anan is a real number for all values of a.
Domain of a Radical Function When the index of the radical is even, the radicand must be greater than or equal to zero. When the index of the radical is odd, the radicand can be any real number.
1. ਨਕਾਰਾਤਮਕ ਸੰਖਿਆ ਦਾ ਵਰਗਮੂਲ ਜੇ b ਇੱਕ ਧਨਾਤਮਕ ਵਾਸਤਵਿਕ ਸੰਖਿਆ ਹੈ, ਤਾਂ −b=bi−b=bi a+bia+bi b=0b=0 a+0·iaa+0·ia ਵਾਸਤਵਿਕ ਸੰਖਿਆ b≠0b≠0 a+bia+bi ਕਾਲਪਨਿਕ ਸੰਖਿਆ a=0a=0 0+bibi0+bibi ਸ਼ੁੱਧ ਕਾਲਪਨਿਕ ਸੰਖਿਆ ਸਾਰਣੀ 8.1 ਇੱਕ ਜਟਿਲ ਸੰਖਿਆ ਨੂੰ ਮਿਆਰੀ ਰੂਪ ਵਿੱਚ ਲਿਖਿਆ ਜਾਂਦਾ ਹੈ ਜਦੋਂ ਇਸਨੂੰ a + bi ਦੇ ਰੂਪ ਵਿੱਚ ਲਿਖਿਆ ਜਾਂਦਾ ਹੈ, ਜਿੱਥੇ a, b ਵਾਸਤਵਿਕ ਸੰਖਿਆਵਾਂ ਹਨ।
2. ਜਟਿਲ ਸੰਯੁਗਮੀਆਂ ਦਾ ਗੁਣਨਫਲ ਜੇ a, b ਵਾਸਤਵਿਕ ਸੰਖਿਆਵਾਂ ਹਨ, ਤਾਂ (a−bi)(a+bi)=a2+b2(a−bi)(a+bi)=a2+b2
3. ਜਟਿਲ ਸੰਖਿਆਵਾਂ ਨੂੰ ਕਿਵੇਂ ਭਾਗ ਕਰਨਾ ਹੈ ਕਦਮ 1. ਅੰਸ਼ ਅਤੇ ਹਰ ਦੋਵਾਂ ਨੂੰ ਮਿਆਰੀ ਰੂਪ ਵਿੱਚ ਲਿਖੋ। ਕਦਮ 2. ਅੰਸ਼ ਅਤੇ ਹਰ ਨੂੰ ਹਰ ਦੇ ਜਟਿਲ ਸੰਯੁਗਮੀ ਨਾਲ ਗੁਣਾ ਕਰੋ। ਕਦਮ 3. ਸਰਲ ਕਰੋ ਅਤੇ ਨਤੀਜੇ ਨੂੰ ਮਿਆਰੀ ਰੂਪ ਵਿੱਚ ਲਿਖੋ।