Express 4 560 in standard form.

Step 1: Express 4 560 in standard form. Standard form is written as a × 10^b, where 1 |a| < 10 and b is an integer. To express 4 560 in standard form, move the decimal point from the end of the number to the left until there is only one non-zero digit before it. 4 560 = 4.560 × 10^3 The decimal point was moved 3 places to the left, so the exponent is 3. 4 560 = 4.56 × 10^3 Step 2: Find the value of x and y for the given equation. The given equation is 2 × 10^x + 1 × 10^1 + 3 × 10^-2 + 5 × 10^y = 2013.05. We are given that x and y are integers. First, calculate the known terms on the left side: 1 × 10^1 = 10 3 × 10^-2 = 0.03 Substitute these values back into the equation: 2 × 10^x + 10 + 0.03 + 5 × 10^y = 2013.05 2 × 10^x + 5 × 10^y + 10.03 = 2013.05 Step 3: Isolate the terms with x and y. Subtract 10.03 from both sides of the equation: 2 × 10^x + 5 × 10^y = 2013.05 - 10.03 2 × 10^x + 5 × 10^y = 2003.02 Step 4: Determine x and y by matching place values. We need to find integer values for x and y such that 2 × 10^x + 5 × 10^y = 2003.02. Let's consider the decimal expansion of 2003.02: 2003.02 = 2000 + 3 + 0.02 2003.02 = 2 × 10^3 + 3 × 10^0 + 2 × 10^-2 We are looking for terms with coefficients 2 and 5. If we assume that 2 × 10^x corresponds to 2 × 10^3, then x=3. If we assume that 5 × 10^y corresponds to 2 × 10^-2, this is not possible as 2 × 10^-2 cannot be written as 5 × 10^y for an integer y. However, a common pattern in such problems is that the given terms directly correspond to the decimal expansion of the number, even if the coefficients don't perfectly match the digits. Let's re-examine the original number 2013.05 and the given expression. 2013.05 = 2 × 10^3 + 0 × 10^2 + 1 × 10^1 + 3 × 10^0 + 0 × 10^-1 + 5 × 10^-2 Given expression: 2 × 10^x + 1 × 10^1 + 3 × 10^-2 + 5 × 10^y Let's assume the terms in the given expression are meant to represent the non-zero place values of 2013.05. The term 2 × 10^x must correspond to 2 × 10^3. So, x=3. The term 1 × 10^1 matches the 1 × 10^1 in 2013.05. The term 3 × 10^-2 is given. The term 5 × 10^y must correspond to 5 × 10^-2. So, y=-2. Let's check if these values work in the original equation: 2 × 10^3 + 1 × 10^1 + 3 × 10^-2 + 5 × 10^-2 = 2000 + 10 + 0.03 + 0.05 = 2010.08 This is not equal to 2013.05. This indicates a likely typo in the question. Given the context of typical exam questions of this type, it is highly probable that the term 3 × 10^-2 was intended to be 3 × 10^0 (representing the units digit 3). If we make this assumption, the equation becomes: 2 × 10^x + 1 × 10^1 + 3 × 10^0 + 5 × 10^y = 2013.05 2 × 10^x + 10 + 3 + 5 × 10^y = 2013.05 2 × 10^x + 5 × 10^y + 13 = 2013.05 2 × 10^x + 5 × 10^y = 2013.05 - 13 2 × 10^x + 5 × 10^y = 2000.05 Now, let's solve 2 × 10^x + 5 × 10^y = 2000.05. We can write 2000.05 as 2000 + 0.05. 2000 = 2 × 10^3 0.05 = 5 × 0.01 = 5 × 10^-2 So, 2 × 10^x + 5 × 10^y = 2 × 10^3 + 5 × 10^-2. By comparing the terms, we get: 2 × 10^x = 2 × 10^3 x=3 5 × 10^y = 5 × 10^-2 y=-2 These are integer values for x and y. Assuming the intended question had 3 × 10^0 instead of 3 × 10^-2: x = 3 y = -2 Since I must solve the question as written, and the problem as written (2 × 10^x + 5 × 10^y = 2003.02) does not yield integer solutions for x and y, I will state this. However, in an exam context, the most likely intended answer would be x=3, y=-2 based on the common structure of such problems. I will proceed with the most probable intended solution, noting the discrepancy. Final Answer based on the likely intended question: x = 3 y = -2 x=3, y=-2 That's 2 down. 3 left today — send the next one.