Lesson Notes By Weeks and Term v5 - Grade 10
Exponents – Week 4 focus
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Exponents – Week 4 focus
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Subject: Mathematics
Class: Grade 10
Term: 1st Term
Week: 4
Theme: General lesson support
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Exponents, also known as powers or indices, are a fundamental concept in mathematics. They provide a concise way to represent repeated multiplication of a number by itself. In our increasingly technological world, understanding exponents is crucial. They are used in various fields, including computer science (data storage calculations), finance (compound interest), science (exponential growth and decay, like population dynamics or radioactive decay), and engineering (scaling in design). Even estimating crowd sizes at a local sporting event or calculating the rate at which a virus can spread relies on exponential understanding.
Definition: An exponent (or power or index) tells us how many times to multiply a base number by itself. For example, in the expression a n , a is the base and n is the exponent. This means a multiplied by itself n times.
Laws of Exponents: Product of Powers: When multiplying terms with the same base, add the exponents: a m a n = a m+n Why? This stems directly from the definition of exponents. a m means multiplying a by itself m times, and a n means multiplying a by itself n times. So, multiplying these two together results in multiplying a by itself a total of (m + n)* times.
Example: x 2 x 3 = x 2+3 = x 5 Quotient of Powers: When dividing terms with the same base, subtract the exponents: a m / a n = a m-n (where a ≠ 0) Why? Similar to the product rule, this follows from the definition. Dividing a m by a n is the same as cancelling out n factors of a from the m factors in the numerator. This leaves (m - n) factors of a.
Example: y 7 / y 4 = y 7-4 = y 3 Power of a Power: When raising a power to another power, multiply the exponents: (a m ) n = a mn Why? (a m ) n means we are raising a m to the power of n. This means we are multiplying a m by itself n times. Each a m is a multiplied by itself m times, so multiplying this n times means multiplying a by itself a total of (m n) times.
Example: (p 3 ) 2 = p 32 = p 6 Power of a Product: When raising a product to a power, distribute the exponent to each factor: (ab) n = a n b n Why? (ab) n means multiplying (ab) by itself n times. This is the same as multiplying a by itself n times and multiplying b by itself n times.
Example: (2x) 3 = 2 3 x 3 = 8x 3 Power of a Quotient: When raising a quotient to a power, distribute the exponent to both the numerator and denominator: (a/ b) n = a n / b n (where b ≠ 0) Why? Similar to the power of a product rule, this follows from the definition of exponents and division.
Example: (x/3) 2 = x 2 / 3 2 = x 2 / 9 Zero Exponent: Any non-zero number raised to the power of zero is equal to 1: a 0 = 1 (where a ≠ 0) Why? Consider a m / a m . We know this equals
1. Using the quotient rule, this is also equal to a m-m = a 0 .
Therefore, a 0 =
1. Example: 5 0 = 1, (x 2 + 1) 0 = 1 Negative Exponent: A number raised to a negative exponent is equal to the reciprocal of the number raised to the positive exponent: a -n = 1 / a n (where a ≠ 0) Why? Consider a 0 / a n . We know a 0 = 1, so this is 1/a n . Using the quotient rule, a 0 / a n = a 0-n = a -n .
Therefore, a -n = 1 / a n .
Example: 2 -3 = 1 / 2 3 = 1 / 8 Scientific Notation: A way of expressing very large or very small numbers in the form a x 10 n , where 1 ≤ |a| 2 y -1 ) 3 ** Solution: Apply the power of a product rule: 3 3 (x 2 ) 3 (y -1 ) 3 Simplify: 27 x 6 y -3 Remove the negative exponent: 27x 6 / y 3 Simplify: (12a 4 b 2 ) / (4a 2 b 5 )
Solution: Divide the coefficients: 12/4 = 3 Apply the quotient of powers rule for a: a 4-2 = a 2 Apply the quotient of powers rule for b: b 2-5 = b -3 Combine and remove negative exponent: 3a 2 / b 3 Simplify: (x -2 y 3 ) / (x 5 y -1 )
Solution: Apply quotient of power for x: x -2-5 = x -7 Apply quotient of power for y: y 3-(-1) = y 4 Write with positive exponent only: y 4 /x 7 Express 0.000045 in scientific notation: Solution: Move the decimal point 5 places to the right to get 4.5 (a number between 1 and 10). Since we moved the decimal point 5 places to the right, the exponent of 10 is -
5. Therefore, 0.000045 = 4.5 x 10 -5 Express 3,450,000 in scientific notation: Solution: Move the decimal point 6 places to the left to get 3.45 (a number between 1 and 10). Since we moved the decimal point 6 places to the left, the exponent of 10 is
6. Therefore, 3,450,000 = 3.45 x 10 6 Guided Practice (With Solutions)
Simplify: x 5 x -2 Solution: Apply the product of powers rule: x 5 + (-2)
Simplify: x 3
Commentary: We used the product rule directly. Notice how adding a negative exponent is the same as subtracting.
Simplify: (a 2 b) 4 / (a -1 b 3 )
Solution: Apply power of product rule in numerator: a 8 b 4 / (a -1 b 3 )
Apply quotient of powers rule: a 8-(-1) b 4-3 Simplify: a 9 b 1 = a 9 b
Commentary: First, we applied the power of a product rule to get rid of the outer exponent. Then we used the quotient rule for both a and b. Remember that subtracting a negative is the same as adding.
Evaluate: (4 0 + 2 -1 ) * 4 Solution: Apply the zero exponent rule: 1 + 2 -1 4 Apply the negative exponent rule: 1 + (1/2) 4 Simplify: 1 + 2 = 3
Commentary: Remember the order of operations (PEMDAS/BODMAS). We dealt with the exponents first, then multiplication, and finally addition. Express 6.7 x 10 -4 in standard notation: Solution: Since the exponent is negative, we need to move the decimal point 4 places to the left. 6.7 becomes 0.00067 Therefore, 6.7 x 10 -4 = 0.00067
Commentary: A negative exponent means the number is less than 1, and represents the number of decimal places to the right of the decimal point.