Problem 5 Raising a Number in Scientific Notation to a Power

Aircraft The expression 1 2 m v 2 1 half . m v squared  gives the kinetic energy, in joules, of an object with a mass of m kg traveling at a speed of v meters per second. What is the kinetic energy of an experimental unmanned jet with a mass of 1.3 × 10 3 1.3 times , 10 cubed  kg traveling at a speed of about 3.1 × 10 3 3.1 times , 10 cubed  m/s?

1 2 m v 2 = 1 2 · ( 1.3 × 10 3 ) ( 3.1 × 10 3 ) 2 Substitute the values for  m  and  v  into the expression.   = 1 2 · 1.3 · 10 3 · 3.1 2 · ( 10 3 ) 2 Raise the two factors to the second power.   =   1 2 · 1.3 · 10 3 · 3 . 1 2 · 10 6 Multiply the exponents of a power raised to a power.   =   1 2 · 1.3 · 3 . 1 2 · 10 3 · 10 6 Use the Commutative Property of Multiplication.   = 1 2 · 1.3 · 3.1 2 · 10 3 + 6 Add exponents of powers with the same base.   = 6.2465 × 10 9 Simplify. Write in scientific notation. table with 6 rows and 3 columns , row1 column 1 , 1 half . m v squared , column 2 equals , 1 half , middle dot . open . 1.3 times , 10 cubed . close . open . 3.1 times , 10 cubed . close squared , column 3 cap substitutethevaluesfor . m , and , v . intotheexpression. , row2 column 1 , , column 2 equals , 1 half , middle dot 1.3 middle dot , 10 cubed , middle dot , 3.1 squared , middle dot . open , 10 cubed , close squared , column 3 cap raisethetwofactorstothesecondpower. , row3 column 1 , , column 2 equals , 1 half . middle dot 1.3 middle dot , 10 cubed , middle dot 3 . , 1 squared , middle dot , 10 to the sixth , column 3 cap multiplytheexponentsofapowerraisedtoapower. , row4 column 1 , , column 2 equals , 1 half . middle dot 1.3 middle dot 3 . , 1 squared , middle dot , 10 cubed , middle dot , 10 to the sixth , column 3 cap usethecap commutativecap propertyofcap multiplication. , row5 column 1 , , column 2 equals , 1 half , middle dot 1.3 middle dot , 3.1 squared , middle dot . 10 super 3 plus 6 end super , column 3 cap addexponentsofpowerswiththesamebase. , row6 column 1 , , column 2 equals , 6.2465 , times , 10 to the ninth , column 3 cap simplify.cap writeinscientificnotation. , end table

The aircraft has a kinetic energy of about 6.2 × 10 9 6.2 times , 10 to the ninth  joules.

✓ Got It?

5. What is the kinetic energy of an aircraft with a mass of 2.5 × 10 5 2.5 times , 10 to the fifth  kg traveling at a speed of 3 × 10 2 m / s ? 3 times , 10 squared , m slash s question mark

✓ Lesson Check

Do you know HOW?

Simplify each expression.

1. ( n 3 ) 6 open , n cubed , close to the sixth
2. ( b − 7 ) 3 open , b super negative 7 end super , close cubed
3. ( 3 a ) 4 open 3 eh close to the fourth
4. ( 9 x 5 ) 2 ( x 2 ) 5 open , 9 , x to the fifth , close squared . open , x squared , close to the fifth

Simplify each expression. Write each answer in scientific notation.

5. ( 4 × 10 5 ) 2 open . 4 times , 10 to the fifth . close squared
6. ( 2 × 10 − 3 ) 5 open . 2 times , 10 super negative 3 end super . close to the fifth

Do you UNDERSTAND?

7. Vocabulary Compare and contrast the property for raising a power to a power and the property for multiplying powers with the same base.
8. Error Analysis One student simplified x 5 + x 5 x to the fifth , plus , x to the fifth  to x 10 . x to the tenth , .  A second student simplified x 5 + x 5 x to the fifth , plus , x to the fifth  to 2 x 5 . 2 , x to the fifth , .  Which student is correct? Explain.
9. Open-Ended Write four different expressions that are equivalent to ( x 4 ) 3 . open , x to the fourth , close cubed . .