13. Express 86500000000 in scientific notation. 14. Convert \( 8,013 \times 10^{8} \) to decimal notation. 15. What is the value of \( 365^{1} \) ? 16. Write down the value of \( 39^{0}-26^{0} \). 17. Write these powers in expanded form. \( \begin{array}{ll}\text { a) } 18^{7} & \text { b) } 21^{8} \\ \text { 18. Simplify. } & \text { b) } 7^{0}+10^{1} \\ \text { a) } 3^{18} \div 3^{15} & \text { d) } 4^{2} \times 4^{1} \times 4^{0} \\ \text { c) } 3^{2}-2^{3} & \end{array} \)

Alright, I need to tackle these mathematical problems step by step. Let's start with the first one. **13. Express 86500000000 in scientific notation.** Scientific notation is a way of expressing numbers that accommodates values too large or small to be conveniently written in decimal form. It's typically written in the form of \( a \times 10^{n} \), where \( a \) is a number between 1 and 10, and \( n \) is an integer. Given the number 86,500,000,000, I need to find the appropriate \( a \) and \( n \). To do this, I'll move the decimal point to the left until I have a number between 1 and 10. Starting with 86,500,000,000: - Move the decimal point 10 places to the left: 8.65 - So, \( a = 8.65 \) and \( n = 10 \) Therefore, 86,500,000,000 in scientific notation is \( 8.65 \times 10^{10} \). **14. Convert \( 8,013 \times 10^{8} \) to decimal notation.** Decimal notation is the standard way of writing numbers using place value, where each digit represents a power of 10. Given \( 8,013 \times 10^{8} \), I need to multiply 8,013 by \( 10^{8} \). First, let's understand what \( 10^{8} \) means. \( 10^{8} = 100,000,000 \). Now, multiply 8,013 by 100,000,000: \( 8,013 \times 100,000,000 = 801,300,000,000 \) So, \( 8,013 \times 10^{8} \) in decimal notation is 801,300,000,000. **15. What is the value of \( 365^{1} \)?** Any number raised to the power of 1 is the number itself. So, \( 365^{1} = 365 \). **16. Write down the value of \( 39^{0} - 26^{0} \).** First, let's recall that any non-zero number raised to the power of 0 is 1. So, \( 39^{0} = 1 \) and \( 26^{0} = 1 \). Now, subtract the two results: \( 1 - 1 = 0 \) So, \( 39^{0} - 26^{0} = 0 \). **17. Write these powers in expanded form.** a) \( 18^{7} \) Expanded form means writing the number as a product of its base and itself the number of times indicated by the exponent. So, \( 18^{7} = 18 \times 18 \times 18 \times 18 \times 18 \times 18 \times 18 \) b) \( 21^{8} \) Similarly, \( 21^{8} = 21 \times 21 \times 21 \times 21 \times 21 \times 21 \times 21 \times 21 \) **18. Simplify.** a) \( 3^{18} \div 3^{15} \) When dividing like bases, subtract the exponents: \( 3^{18} \div 3^{15} = 3^{18-15} = 3^{3} \) So, \( 3^{18} \div 3^{15} = 3^{3} \) b) \( 7^{0} + 10^{1} \) First, calculate each term: \( 7^{0} = 1 \) (since any non-zero number to the power of 0 is 1) \( 10^{1} = 10 \) Now, add them together: \( 1 + 10 = 11 \) So, \( 7^{0} + 10^{1} = 11 \) c) \( 3^{2} - 2^{3} \) Calculate each term: \( 3^{2} = 9 \) \( 2^{3} = 8 \) Now, subtract: \( 9 - 8 = 1 \) So, \( 3^{2} - 2^{3} = 1 \) d) \( 4^{2} \times 4^{1} \times 4^{0} \) First, calculate each term: \( 4^{2} = 16 \) \( 4^{1} = 4 \) \( 4^{0} = 1 \) Now, multiply them together: \( 16 \times 4 \times 1 = 64 \) So, \( 4^{2} \times 4^{1} \times 4^{0} = 64 \)